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Binomial Test Calculator

This binomial test calculator determines the probability of a particular outcome (K) across a certain number of trials ( n ), where there are precisely two possible outcomes.

To use the calculator, enter the values of n , K and p into the table below ( q will be calculated automatically), where n is the number of trials or observations, K is number of occasions the actual (or stipulated) outcome occurred, and p is the probability the outcome will occur on any particular occasion.

Things to remember: (a) the binomial test is appropriate only when you've got just two possible outcomes (or categories, etc.); (b) n and K will be frequencies; and (c) the value for p will fall somewhere between 0 and 1 - it's a proportion.

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Binomial Probability Calculator

Use the Binomial Calculator to compute individual and cumulative binomial probabilities. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems .

To learn more about the binomial distribution, go to Stat Trek's tutorial on the binomial distribution .

• Enter a value in each of the first three text boxes (the unshaded boxes).
• Click the Calculate button to compute binomial and cumulative probabilities.

Frequently-Asked Questions

Instructions: To find the answer to a frequently-asked question, simply click on the question.

What is a binomial experiment?

A binomial experiment has the following characteristics:

• The experiment involves repeated trials.
• Each trial has only two possible outcomes - a success or a failure.
• The probability that any trial will result in success is constant.
• All of the trials in the experiment are independent.

A series of coin tosses is a perfect example of a binomial experiment. Suppose we toss a coin three times. Each coin flip represents a trial, so this experiment would have 3 trials. Each coin flip also has only two possible outcomes - a Head or a Tail. We could call a Head a success; and a Tail, a failure. The probability of a success on any given coin flip would be constant (i.e., 50%). And finally, the outcome on any coin flip is not affected by previous or succeeding coin flips; so the trials in the experiment are independent.

What is a binomial distribution?

A binomial distribution is a probability distribution . It refers to the probabilities associated with the number of successes in a binomial experiment .

For example, suppose we toss a coin three times and suppose we define Heads as a success. This binomial experiment has four possible outcomes: 0 Heads, 1 Head, 2 Heads, or 3 Heads. The probabilities associated with each possible outcome are an example of a binomial distribution , as shown below.

What is the number of trials?

The number of trials refers to the number of replications in a binomial experiment.

Suppose that we conduct the following binomial experiment. We flip a coin and count the number of Heads. We classify Heads as success; tails, as failure. If we flip the coin 3 times, then 3 is the number of trials. If we flip it 20 times, then 20 is the number of trials.

Note: Each trial results in a success or a failure. So the number of trials in a binomial experiment is equal to the number of successes plus the number of failures.

What is the number of successes?

Each trial in a binomial experiment can have one of two outcomes. The experimenter classifies one outcome as a success; and the other, as a failure. The number of successes in a binomial experient is the number of trials that result in an outcome classified as a success.

What is the probability of success on a single trial?

In a binomial experiment, the probability of success on any individual trial is constant. For example, the probability of getting Heads on a single coin flip is always 0.50. If "getting Heads" is defined as success, the probability of success on a single trial would be 0.50.

What is the binomial probability?

A binomial probability refers to the probability of getting EXACTLY r successes in a specific number of trials. For instance, we might ask: What is the probability of getting EXACTLY 2 Heads in 3 coin tosses. That probability (0.375) would be an example of a binomial probability.

In a binomial experiment, the probability that the experiment results in exactly x successes is indicated by the following notation: P(X=x);

What is the cumulative binomial probability?

Cumulative binomial probability refers to the probability that the value of a binomial random variable falls within a specified range.

The probability of getting AT MOST 2 Heads in 3 coin tosses is an example of a cumulative probability. It is equal to the probability of getting 0 heads (0.125) plus the probability of getting 1 head (0.375) plus the probability of getting 2 heads (0.375). Thus, the cumulative probability of getting AT MOST 2 Heads in 3 coin tosses is equal to 0.875.

Notation associated with cumulative binomial probability is best explained through illustration. The probability of getting FEWER THAN 2 successes is indicated by P(X<2); the probability of getting AT MOST 2 successes is indicated by P(X≤2); the probability of getting AT LEAST 2 successes is indicated by P(X≥2); the probability of getting MORE THAN 2 successes is indicated by P(X>2).

Sample Problem

• The probability of success (i.e., getting a Head) on any single trial is 0.5.
• The number of trials is 12.
• The number of successes is 7 (since we define getting a Head as success).

Therefore, we plug those numbers into the Binomial Calculator and hit the Calculate button.

The calculator reports that the binomial probability is 0.193. That is the probability of getting EXACTLY 7 Heads in 12 coin tosses. (The calculator also reports the cumulative probabilities. For example, the probability of getting AT MOST 7 heads in 12 coin tosses is a cumulative probability equal to 0.806.)

• The probability of success for any individual student is 0.6.
• The number of trials is 3 (because we have 3 students).
• The number of successes is 2.

The calculator reports that the probability that two or fewer of these three students will graduate is 0.784.

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Binomial Test Calculator

Calculate Binomial Test with this free online tool

How to Calculate Binomial Test

• Set values for Trials, Successes, P(success), alpha, and alternative.
• View the result.
• If there is an error in red text, then try changing the inputs. If it persists, then click the Feedback button at the top of the page.

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Binomial Distribution Calculator

Use this binomial probability calculator to easily calculate binomial cumulative distribution function and probability mass given the probability on a single trial, the number of trials and events. It can calculate the probability of success if the outcome is a binomial random variable, for example if flipping a coin. The calculator can also solve for the number of trials required.

Related calculators

• Using the Binomial Probability Calculator
• What is a Binomial Distribution?
• What is a Binomial Probability?
• Binomial Cumulative Distribution Function (CDF)

    Using the Binomial Probability Calculator

You can use this tool to solve either for the exact probability of observing exactly x events in n trials, or the cumulative probability of observing X ≤ x , or the cumulative probabilities of observing X < x or X ≥ x or X > x. Simply enter the probability of observing an event (outcome of interest, success) on a single trial (e.g. as 0.5 or 1/2, 1/6 and so on), the number of trials and the number of events you want the probability calculated for. As long as the procedure generating the event conforms to the random variable model under a Binomial distribution the calculator applies. In other words, X must be a random variable generated by a process which results in Binomially-distributed, Independent and Identically Distributed outcomes (BiIID). For example, you can compute the probability of observing exactly 5 heads from 10 coin tosses of a fair coin (24.61%) , of rolling more than 2 sixes in a series of 20 dice rolls (67.13%) and so on. See more examples below.

Under the same conditions you can use the binomial probability distribution calculator above to compute the number of attempts you would need to see x or more outcomes of interest (successes, events). For example, if you know you have a 1% chance (1 in 100) to get a prize on each draw of a lottery, you can compute how many draws you need to participate in to be 99.99% certain you win at least 1 prize ( 917 draws ). Note that this example doesn't apply if you are buying tickets for a single lottery draw (the events are not independent).

Sequences of Bernoulli trials: trials in which the outcome is either 1 or 0 with the same probability on each trial result in and are modelled as binomial distribution so any such problem is one which can be solved using the above tool: it essentially doubles as a coin flip calculator .

    What is a Binomial Distribution?

The binomial distribution X~Bin(n,p) is a probability distribution which results from the number of events in a sequence of n independent experiments with a binary / Boolean outcome: true or false, yes or no, event or no event, success or failure. These are also known as Bernoulli trials and thus a Binomial distribution is the result of a sequence of Bernoulli trials. The parameters which describe it are n - number of independent experiments and p the probability of an event of interest in a single experiment. It is often used as a teaching device and the practical applications of probability theory and statistics due its many desirable properties such as a known standard deviation and easy to compute cumulative distribution function and inverse function.

The binomial probability distribution can be used to model the number of events in a sample of size n drawn with replacement from a population of size N, e.g. coin tosses, dice rolls, and so on. If the sampling is carried out without replacement they are no longer independent and the result is a hypergeometric distribution, although the binomial remains a decent approximation if N >> n.

The above is a randomly generated binomial distribution from 10,000 simulated binomial experiments, each with 10 Bernoulli trials with probability of observing an event of 0.2 (20%).

    What is a Binomial Probability?

A probability for a certain outcome from a binomial distribution is what is usually referred to as a "binomial probability". It can be calculated using the formula for the binomial probability distribution function (PDF), a.k.a. probability mass function (PMF): f(x) , as follows:

Note that the above equation is for the probability of observing exactly the specified outcome. However, often when searching for a binomial probability formula calculator people are actually looking to calculate the cumulative probability of a binomially-distributed random variable: the probability of observing x or less than x events (successes, outcomes of interest).

    Binomial Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of the Binomial distribution is what is needed when you need to compute the probability of observing less than or more than a certain number of events/outcomes/successes from a number of trials. The Binomial CDF formula is simple:

Therefore, the cumulative binomial probability is simply the sum of the probabilities for all events from 0 to x. Our binomial distribution calculator uses the formula above to calculate the cumulative probability of events less than or equal to x, less than x, greater than or equal to x and greater than x for you. These are all cumulative binomial probabilities.

The inverse function is required when computing the number of trials required to observe a certain number of events, or more, with a certain probability. For this we use the inverse normal distribution function which provides a good enough approximation.

    Examples

Example 1: Coin flipping. If a fair coin (p = 1/2 = 0.5) is tossed 100 times, what is the probability of observing exactly 50 heads? What is the probability of observing more than 50 heads? Entering 0.5 or 1/2 in the calculator and 100 for the number of trials and 50 for "Number of events" we get that the chance of seeing exactly 50 heads is just under 8% while the probability of observing more than 50 is a whopping 46%. While in an infinite number of coin flips a fair coin will tend to come up heads exactly 50% of the time, in any small number of flips it is highly unlikely to observe exactly 50% heads.

Example 2: Dice rolling. If a fair dice is thrown 10 times, what is the probability of throwing at least one six? We know that a dice has six sides so the probability of success in a single throw is 1/6. Thus, using n=10 and x=1 we can compute using the Binomial CDF that the chance of throwing at least one six (X ≥ 1) is 0.8385 or 83.85 percent.

Cite this calculator & page

If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Binomial Distribution Calculator" , [online] Available at: https://www.gigacalculator.com/calculators/binomial-probability-calculator.php URL [Accessed Date: 15 May, 2024].

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     Statistical calculators

Hypothesis Testing Calculator

Related: confidence interval calculator, type ii error.

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

Descriptive Statistics

Hypothesis test, binomial test calculator.

The domain of statistics is vast, with numerous tests and tools designed to decode the complexities of data. Among these, the binomial test stands out as a powerful technique to test the probability of observed outcomes. Whether you're examining the success rate of a new drug, the likelihood of customers choosing a product, or any scenario involving two potential outcomes, the Binomial Test Calculator is your analytical companion. This post delves into the significance of the binomial test and how our calculator simplifies its execution.

Understanding the Binomial Test

At its core, the binomial test determines whether the observed proportion of successes in a sample matches a pre-defined expectation. It's especially handy when you're dealing with yes/no, success/failure, or any other dichotomous outcomes.

Key Features of the Binomial Test Calculator

• User-Friendly Interface: Regardless of your statistical prowess, our calculator is designed for ease of use, ensuring every user can quickly input data and derive results.
• Quick Analysis: The need for manual calculations or complex software setups is eliminated. Input your data, and obtain results promptly.
• Detailed Outputs: Along with the p-value, the calculator provides clear interpretations, helping you gauge whether the observed outcomes significantly deviate from expectations.
• Graphical Illustration: Visual learners can benefit from the distribution graphs, making it easier to understand the data's significance in a graphical context.

How to Navigate the Binomial Test Calculator

• Data Input: Begin by inputting the number of trials, the number of observed successes, and the expected probability of success.
• Calculation: After feeding the data, click 'Calculate'. The calculator promptly processes the data and provides the results.
• Interpreting Results: A crucial metric provided is the p-value. Typically, a p-value less than 0.05 indicates that the observed data is significantly different from what was expected under the null hypothesis.

Practical Applications

The binomial test and our calculator find usage in various fields:

• Medical Research: To analyze the effectiveness of a treatment when compared to a known success rate.
• Marketing Studies: To determine if a new advertisement has a significant impact on product choices.
• Quality Control: To assess if the defect rate in a production batch deviates from acceptable standards.

The Binomial Test Calculator transforms these analytical tasks into smooth operations, ensuring accuracy and efficiency.

In Conclusion

The realm of statistics offers profound insights, provided we have the right tools in our arsenal. The Binomial Test Calculator serves as a bridge between complex statistical principles and actionable insights. Whether you're a researcher, a student, or a professional looking to make data-driven decisions, our calculator ensures you do so with confidence and clarity. Dive into the fascinating world of binomial tests and let our calculator guide your explorations!

Binomial Distribution Calculator

What is the binomial probability, binomial probability formula, how to use the binomial distribution calculator: an example, how to calculate cumulative probabilities, binomial probability distribution experiments, mean and variance of binomial distribution, other considerations.

This binomial distribution calculator is here to help you with probability problems in the following form: what is the probability of a certain number of successes in a sequence of events? Read on to learn what exactly is the binomial probability distribution, when and how to apply it, and learn the binomial probability formula. Find out what is binomial distribution, and discover how binomial experiments are used in various settings.

Imagine you're playing a game of dice. To win, you need exactly three out of five dice to show a result equal to or lower than 4. The remaining two dice need to show a higher number. What is the probability of you winning?

This is a sample problem that can be solved with our binomial probability calculator. You know the number of events (it is equal to the total number of dice, so five); you know the number of successes you need (precisely 3); you also can calculate the probability of one single success occurring (4 out of 6, so 0.667). This is all the data required to find the binomial probability of you winning the game of dice.

Note that to use the binomial distribution calculator effectively, the events you analyze must be independent . It means that all the trials in your example are supposed to be mutually exclusive.

The first trial's success doesn't affect the probability of success or the probability of failure in subsequent events, and they stay precisely the same. In the case of a dice game, these conditions are met: each time you roll a die constitutes an independent event.

Sometimes you may be interested in the number of trials you need to achieve a particular outcome. For instance, you may wonder how many rolls of a die are necessary before you throw a six three times. Such questions may be addressed using a related statistical tool called the negative binomial distribution. Make sure to learn about it with Omni's negative binomial distribution calculator .

Also, you may check our normal approximation to binomial distribution calculator and the related continuity correction calculator.

To find this probability, you need to use the following equation:

P(X=r) = nCr × p r × (1-p) n-r

• n – Total number of events;
• r – Number of required successes;
• p  – Probability of one success;
• nCr – Number of combinations (so-called "n choose r"); and
• P(X=r) – Probability of an exact number of successes happening.

You should note that the result is the probability of an exact number of successes. For example, in our game of dice, we needed precisely three successes – no less, no more. What would happen if we changed the rules so that you need at least three successes? Well, you would have to calculate the probability of exactly three, precisely four, and precisely five successes and sum all of these values together.

Let's solve the problem of the game of dice together.

Determine the number of events. n is equal to 5, as we roll five dice.

Determine the required number of successes. r is equal to 3, as we need exactly three successes to win the game.

The probability of rolling 1, 2, 3, or 4 on a six-sided die is 4 out of 6, or 0.667. Therefore p is equal to 0.667 or 66.7%.

Calculate the number of combinations (5 choose 3). You can use the combination calculator to do it. This number, in our case, is equal to 10.

Substitute all these values into the binomial probability formula above:

P(X = 3) = 10 × 0.667 3 × (1-0.667) (5-3) = 10 × 0.667 3 × (1-0.667) (5-3) = 10 × 0.296 × 0.333 2 = 2.96 × 0.111 = 0.329

You can also save yourself some time and use the binomial distribution calculator instead :)

Sometimes, instead of an exact number of successes, you want to know the probability of getting r or more successes or r or less successes. To calculate the probability of getting any range of successes:

• Use the binomial probability formula to calculate the probability of success (P) for all possible values of r you are interested in.
• Sum the values of P for all r within the range of interest.

For example, the probability of getting two or fewer successes when flipping a coin four times (p = 0.5 and n = 4) would be:

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

P(X ≤ 2) = 37.5% + 25% + 6.25%

P(X ≤ 2) = 68.75%

This calculation is made easy using the options available on the binomial distribution calculator. You can change the settings to calculate the probability of getting:

• Exactly r successes: P(X = r)
• r or more successes: P(X ≥ r)
• r or fewer successes: P(X ≤ r)
• Between r₀ and r₁ successes P(r₀ ≤ X ≤ r₁)

The binomial distribution turns out to be very practical in experimental settings . However, the output of such a random experiment needs to be binary : pass or failure, present or absent, compliance or refusal. It's impossible to use this design when there are three possible outcomes.

At the same time, apart from rolling dice or tossing a coin, it may be employed in somehow less clear cases. Here are a couple of questions you can answer with the binomial probability distribution:

• Will a new drug work on a randomly selected patient?
• Will a light bulb you just bought work properly, or will it be broken?
• What is a chance of correctly answering a test question you just drew?
• What is a probability of a random voter to vote for a candidate in an election?
• How likely is it for a group of students to be accepted to a prestigious college?

Experiments with precisely two possible outcomes, such as the ones above, are typical binomial distribution examples, often called the Bernoulli trials .

In practice, you can often find the binomial probability examples in fields like quality control , where this method is used to test the efficiency of production processes. The inspection process based on the binomial distribution is designed to perform a sufficient number of checkups and minimize the chances of manufacturing a defective product.

If you don't know the probability of an independent event in your experiment ( p ), collect the past data in one of your binomial distribution examples, and divide the number of successes ( y ) by the overall number of events p = y/n .

Once you have determined your rate of success (or failure) in a single event, you need to decide what's your acceptable number of successes (or failures) in the long run. For example, one defective product in a batch of fifty is not a tragedy, but you wouldn't like to have every second product faulty, would you?

Bernoulli trials are also perfect at solving network systems . Interestingly, they may be used to work out paths between two nodes on a diagram. This is the case of the Wheatstone bridge network, a representation of a circuit built for electrical resistance measurement.

Like the binomial distribution table , our calculator produces results that help you assess the chances that you will meet your target. However, if you like, you may take a look at this binomial distribution table . It tells you what is the binomial distribution value for a given probability and number of successes.

One of the most exciting features of binomial distributions is that they represent the sum of a number n of independent events. Each of them ( Z ) may assume the values of 0 or 1 over a given period.

Let's say the probability that each Z occurs is p . Since the events are not correlated, we can use random variables' addition properties to calculate the mean (expected value) of the binomial distribution μ = np .

The variance of a binomial distribution is given as: σ² = np(1-p) . The larger the variance, the greater the fluctuation of a random variable from its mean. A small variance indicates that the results we get are spread out over a narrower range of values.

The standard deviation of binomial distribution, another measure of a probability distribution dispersion, is simply the square root of the variance, σ . Keep in mind that the standard deviation calculated from your sample (the observations you actually gather) may differ from the entire population's standard deviation. If you find this distinction confusing, there here's a great explanation of this distinction .

There's a clear-cut intuition behind these formulas. Suppose this time that I flip a coin 20 times:

• My p is then equal to 0.5 (unless, of course, the coin is rigged);
• Each Z has an equivalent chance of 0 or 1;
• The number of trials, n , is 20.

This sequence of events fulfills the prerequisites of a binomial distribution.

The mean value of this simple experiment is: np = 20 × 0.5 = 10 . We can say that on average if we repeat the experiment many times, we should expect heads to appear ten times.

The variance of this binomial distribution is equal to np(1-p) = 20 × 0.5 × (1-0.5) = 5 . Take the square root of the variance, and you get the standard deviation of the binomial distribution, 2.24 . Accordingly, the typical results of such an experiment will deviate from its mean value by around 2. Hence, in most of the trials, we expect to get anywhere from 8 to 12 successes.

Use our binomial probability calculator to get the mean, variance, and standard deviation of binomial distribution based on the number of events you provided and the probability of one success.

Developed by a Swiss mathematician Jacob Bernoulli , the binomial distribution is a more general formulation of the Poisson distribution. In the latter, we simply assume that the number of events (trials) is enormous, but the probability of a single success is small.

The binomial distribution is closely related to the binomial theorem , which proves to be useful for computing permutations and combinations. Make sure to check out our permutations calculator , too!

Keep in mind that the binomial distribution formula describes a discrete distribution . The possible outcomes of all the trials must be distinct and non-overlapping. What's more, the two outcomes of an event must be complementary: for a given p , there's always an event of q = 1-p .

If there's a chance of getting a result between the two, such as 0.5, the binomial distribution formula should not be used. The same goes for the outcomes that are non-binary, e.g., an effect in your experiment may be classified as low, moderate, or high.

However, for a sufficiently large number of trials, the binomial distribution formula may be approximated by the Gaussian (normal) distribution specification, with a given mean and variance. That allows us to perform the so-called continuity correction , and account for non-integer arguments in the probability function.

Maybe you still need some practice with the binomial probability distribution examples?

Try to solve the dice game's problem again, but this time you need three or more successes to win it. How about the chances of getting exactly 4?

Is the binomial distribution discrete or continuous?

The binomial distribution is discrete – it takes only a finite number of values.

How do I find the mean of a binomial distribution?

To calculate the mean (expected value) of a binomial distribution B(n,p) you need to multiply the number of trials n by the probability of successes p , that is: mean = n × p .

How do I find the standard deviation of a binomial distribution?

To find the standard deviation of a binomial distribution B(n,p) :

• Compute the variance as n × p × (1-p) , where n is the number of trials and p is the probability of successes.
• Take the square root of the number obtained in Step 1.
• That's it! Congrats :)

What is the probability of 3 successes in 5 trials if the probability of success is 0.5?

To find this probability, you need to:

Recall the binomial distribution formula P(X = r) = nCr × pʳ × (1-p)ⁿ⁻ʳ . We'll use it with the following data:

Number of trials: n = 5 ;

Number of successes: r = 3 ; and

Probability of success: p = 0.5 .

Calculate 5 choose 3 : nCr = 10 .

Plug these values into the formula:

P(X = 3) = 10 × 0.5² × 0.5³ = 0.3125 .

The probability you're looking for is 31.25% .

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Binomial Probability Calculator

• Binomial Probability

As we continue our journey through the intricate world of statistics, we encounter a fundamental concept: binomial probability. This foundational idea is a cornerstone of various scientific, economic, and social disciplines, applicable everywhere. Our goal is to help you grasp this concept effectively, and to assist in that endeavor, we present our Binomial Probability Calculator - a user-friendly tool designed to make your calculations as straightforward as possible.

Delving Deeper into Binomial Probability

At its core, binomial probability deals with discrete events happening in a fixed number of independent trials, each having two possible outcomes - success or failure. The classic example of a binomial experiment is flipping a coin - with only two outcomes (heads or tails), and each flip independent from the next.

The mathematical formula that represents binomial probability is as follows:

P(x; n, p) = \binom{n}{x} \cdot p^{x} \cdot (1 - p)^{n-x}

This formula breaks down into:

• P(x; n, p): the probability of x successes in n trials
• C(n, x): the combination function, representing the number of possible combinations of n trials taken x at a time
• p: the probability of success on a single trial
• (1 - p): the probability of failure (often represented as q)

This formula allows us to calculate the probability of obtaining a specific number of successes in a set number of trials.

Unleashing the Potential of the Binomial Probability Calculator

Our calculator is designed to save you the hassle of performing complex calculations manually. It offers a user-friendly interface, allowing you to input your values and receive an accurate, instant result. This precision and speed free up your time and cognitive resources, enabling you to focus on interpreting the results and applying them in your context.

Guidance on Using the Binomial Probability Calculator

The operation of our calculator is simple. Follow these steps to get your result:

• Enter Your Values: You will input three parameters: the number of trials (n), the number of successful outcomes you are calculating the probability for (x), and the probability of success on a single trial (p).
• Review Your Result: The binomial probability will be displayed instantly.

Walking Through an Example: The Binomial Probability Calculator in Action

To demonstrate the utility of our calculator, let's consider a classic example.

Imagine flipping a fair coin (where the probability of landing on heads, p, equals 0.5) ten times. You want to know the probability of getting exactly five heads.

Let's run these values through our calculator:

• Input Values: Enter 10 for the number of trials (n), 5 for the number of successful outcomes (x), and 0.5 for the probability of success (p).
• Review: The calculator promptly computes the binomial probability, providing the likelihood of getting exactly 5 heads in 10 flips.

The Versatility and Applicability of Binomial Probability

Binomial probability finds applications across various domains:

• Finance: Investors often use binomial probability to understand the likelihood of a specific number of investments being successful.
• Quality Control: Manufacturers can use it to predict the number of defective items in a large batch.
• Biology: Geneticists use binomial probability to calculate the chances of organisms inheriting specific traits.
• Sports: Coaches and analysts use it to analyze patterns of win and loss, making strategic decisions based on the results.
• Survey and Poll Analysis: It can help interpret the results of yes/no questions across a sample population.

Our Binomial Probability Calculator is your go-to tool for all these applications and more. Whether you're a student trying to complete your statistics homework, a researcher examining the effects of a new drug, or a market analyst trying to predict future trends, our calculator can facilitate your work and boost your understanding of binomial probability.

Always remember, statistics is not just about number crunching. It's about making sense of patterns, understanding uncertainty, and making informed decisions based on the data at hand. Our Binomial Probability Calculator is a stepping stone on your journey to statistical mastery. Enjoy exploring and happy calculating!

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Sign and binomial test

Use the binomial test when there are two possible outcomes. You know how many of each kind of outcome (traditionally called "success" and "failure") occurred in your experiment. You also have a hypothesis for what the true overall probability of "success" is. The binomial test answers this question: If the true probability of "success" is what your theory predicts, then how likely is it to find results that deviate as far, or further, from the prediction.

Analyze, graph and present your scientific work easily with GraphPad Prism. No coding required.

Binomial test

The binomial test is a hypothesis test used when there is a categorical variable with two expressions, e.g., gender with "male" and "female". The binomial test can then check whether the frequency distribution of the variable corresponds to an expected distribution, e.g.:

• Men and women are equally represented.
• The proportion of women is 54%.

This is a special case when you want to test whether the frequency distribution of the variables is random or not. In this case, the probability of occurrence is set to 50%.

The binomial test can therefore be used to test whether or not the frequency distribution of a sample is the same as that of the population.

The binomial test checks whether the frequency distribution of a variable with two values/categories in the sample corresponds to the distribution in the population.

Hypotheses in binomial test

The hypothesis of the binomial test results in the one tailed case to

• Null hypothesis: The frequency distribution of the sample corresponds to that of the population.
• Alternative hypothesis: The frequency distribution of the sample does not correspond to that of the population.

Thus, the non-directional hypothesis only tests whether there is a difference or not, but not in which direction this difference goes.

In the two sided case, the aim is to investigate whether the probability of occurrence of an expression in the sample is greater or less than a given or true percentage.

In this case, an expression is defined as "success" and it is checked whether the true "probability of success" is smaller or larger than that in the sample.

The alternative hypothesis then results in:

• Alternative hypothesis: True probability of success is smaller/larger than specified value

Binomial test calculation

To calculate a binomial test you need the sample size, the number of cases that are positive of it, and the probability of occurrence in the population.

Binomial test example

A possible example for a binomial test would be the question whether the gender ratio in the specialization marketing at the university XY differs significantly from that of all business students at the university XY (population).

Listed below are the students majoring in marketing; women make up 55% of the total business degree program.

Binomial test with DATAtab:

Calculate the example in the statistics calculator. Simply add the upper table including the first row into the hypothesis test calculator .

DATAtab gives you the following result for this example data:

Interpretation of a Binomial Test

With an expected test value of 55%, the p-value is 0.528. This means that the p-value is above the signification level of 5% and the result is therefore not significant. Consequently, the null hypothesis must not be rejected. In terms of content, this means that the gender ratio of the marketing specialization (=sample) does not differ significantly from that of all business administration students at XY University (=population).

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Cite DATAtab: DATAtab Team (2024). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net

Hypothesis Testing with the Binomial Distribution

Contents Toggle Main Menu 1 Hypothesis Testing 2 Worked Example 3 See Also

Hypothesis Testing

To hypothesis test with the binomial distribution, we must calculate the probability, $p$, of the observed event and any more extreme event happening. We compare this to the level of significance $\alpha$. If $p>\alpha$ then we do not reject the null hypothesis. If $p<\alpha$ we accept the alternative hypothesis.

Worked Example

A coin is tossed twenty times, landing on heads six times. Perform a hypothesis test at a $5$% significance level to see if the coin is biased.

First, we need to write down the null and alternative hypotheses. In this case

The important thing to note here is that we only need a one-tailed test as the alternative hypothesis says “in favour of tails”. A two-tailed test would be the result of an alternative hypothesis saying “The coin is biased”.

We need to calculate more than just the probability that it lands on heads $6$ times. If it landed on heads fewer than $6$ times, that would be even more evidence that the coin is biased in favour of tails. Consequently we need to add up the probability of it landing on heads $1$ time, $2$ times, $\ldots$ all the way up to $6$ times. Although a calculation is possible, it is much quicker to use the cumulative binomial distribution table. This gives $\mathrm{P}[X\leq 6] = 0.058$.

We are asked to perform the test at a $5$% significance level. This means, if there is less than $5$% chance of getting less than or equal to $6$ heads then it is so unlikely that we have sufficient evidence to claim the coin is biased in favour of tails. Now note that our $p$-value $0.058>0.05$ so we do not reject the null hypothesis. We don't have sufficient evidence to claim the coin is biased.

But what if the coin had landed on heads just $5$ times? Again we need to read from the cumulative tables for the binomial distribution which shows $\mathrm{P}[X\leq 5] = 0.021$, so we would have had to reject the null hypothesis and accept the alternative hypothesis. So the point at which we switch from accepting the null hypothesis to rejecting it is when we obtain $5$ heads. This means that $5$ is the critical value .

Selecting a Hypothesis Test

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