What does it mean for an experiment to be binomial?

Binomial probability refers to the probability of exactly x successes on n repeated trials in an experiment which has two possible outcomes (commonly called a binomial experiment).

If the probability of success on an individual trial is p , then the binomial probability is n C x ⋅ p x ⋅ ( 1 − p ) n − x .

Here n C x indicates the number of different combinations of x objects selected from a set of n objects. Some textbooks use the notation ( n x ) instead of n C x .

Note that if p is the probability of success of a single trial, then ( 1 − p ) is the probability of failure of a single trial.

Example:

What is the probability of getting 6 heads, when you toss a coin 10 times?

In a coin-toss experiment, there are two outcomes: heads and tails. Assuming the coin is fair , the probability of getting a head is 1 2 or 0.5 .

The number of repeated trials: n = 10

The number of success trials: x = 6

The probability of success on individual trial: p = 0.5

Use the formula for binomial probability.

10 C 6 ⋅ ( 0.5 ) 6 ⋅ ( 1 − 0.5 ) 10 − 6

Simplify.

≈ 0.205

If the outcomes of the experiment are more than two, but can be broken into two probabilities p and q such that p + q = 1 , the probability of an event can be expressed as binomial probability.

For example, if a six-sided die is rolled 10 times, the binomial probability formula gives the probability of rolling a three on 4 trials and others on the remaining trials.

The experiment has six outcomes. But the probability of rolling a 3 on a single trial is 1 6 and rolling other than 3 is 5 6 . Here, 1 6 + 5 6 = 1 .

The binomial probability is:

10 C 4 ⋅ ( 1 6 ) 4 ⋅ ( 1 − 1 6 ) 10 − 4

Simplify.

≈ 0.054

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To understand binomial distributions and binomial probability, it helps to understand binomial experiments and some associated notation; so we cover those topics first.

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A binomial experiment is a statistical experiment that has the following properties:

• The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

Consider the following statistical experiment. You flip a coin 2 times and count the number of times the coin lands on heads. This is a binomial experiment because:

• The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials.

Notation

The following notation is helpful, when we talk about binomial probability.

• nCr: The number of combinations of n things, taken r at a time.

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A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution.

Suppose we flip a coin two times and count the number of heads (successes). The binomial random variable is the number of heads, which can take on values of 0, 1, or 2. The binomial distribution is presented below.

The binomial distribution has the following properties:

Binomial Formula and Binomial Probability

The binomial probability refers to the probability that a binomial experiment results in exactly x successes. For example, in the above table, we see that the binomial probability of getting exactly one head in two coin flips is 0.50.

Given x, n, and P, we can compute the binomial probability based on the binomial formula:

Binomial Formula. Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success on an individual trial is P, then the binomial probability is:

b(x; n, P) = nCx * Px * (1 - P)n - x or

b(x; n, P) = { n! / [ x! (n - x)! ] } * Px * (1 - P)n - x

Example 1

Suppose a die is tossed 5 times. What is the probability of getting exactly 2 fours?

Solution: This is a binomial experiment in which the number of trials is equal to 5, the number of successes is equal to 2, and the probability of success on a single trial is 1/6 or about 0.167. Therefore, the binomial probability is:

b(2; 5, 0.167) = 5C2 * (0.167)2 * (0.833)3
b(2; 5, 0.167) = 10 * (0.167)2 * (0.833)3
b(2; 5, 0.167) = 0.161

Cumulative Binomial Probability

A cumulative binomial probability refers to the probability that the binomial random variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper limit).

To compute a cumulative binomial probability, we find the sum of relevant individual binomial probabilities, as illustrated below in Example 2.

Example 2

What is the probability of obtaining 45 or fewer heads in 100 tosses of a coin?

Solution: To solve this problem, we compute 46 individual binomial probabilities, using the binomial formula. The sum of all these binomial probabilities is the answer we seek. Thus,

b(x < 45; 100, 0.5) = b(x = 0; 100, 0.5) + b(x = 1; 100, 0.5) + . . . + b(x = 44; 100, 0.5) + b(x = 45; 100, 0.5)
b(x < 45; 100, 0.5) = 0.184

As you may have noticed, the binomial formula requires many time-consuming computations. The Binomial Calculator can do this work for you - quickly, easily, and error-free. Use the Binomial Calculator to compute binomial probabilities and cumulative binomial probabilities. The calculator is free. It can found in the Stat Trek main menu under the Stat Tools tab. Or you can tap the button below.

Example 3

The probability that a student is accepted to a prestigious college is 0.3. If 5 students from the same school apply, what is the probability that at most 2 are accepted?

Solution: To solve this problem, we compute 3 individual probabilities, using the binomial formula. The sum of all these probabilities is the answer we seek. Thus,

b(x < 2; 5, 0.3) = b(x = 0; 5, 0.3) + b(x = 1; 5, 0.3) + b(x = 2; 5, 0.3)
b(x < 2; 5, 0.3) = 0.1681 + 0.3601 + 0.3087
b(x < 2; 5, 0.3) = 0.8369

Example 4

What is the probability that the world series will last 4 games? 5 games? 6 games? 7 games? Assume that the teams are evenly matched.

Solution: The solution to this problem requires a creative application of the binomial formula. If you can follow the logic of this solution, you have a good understanding of the material covered in the tutorial, to this point.

In the world series, there are two baseball teams. The series ends when the winning team wins 4 games. Therefore, we define a success as a win by the team that ultimately becomes the world series champion.

For the purpose of this analysis, we assume that the teams are evenly matched. Therefore, the probability that a particular team wins a particular game is 0.5.

Let's look first at the simplest case. What is the probability that the series lasts only 4 games. This can occur if one team wins the first 4 games. The probability of the National League team winning 4 games in a row is:

b(4; 4, 0.5) = 4C4 * (0.5)4 * (0.5)0 = 0.0625

Similarly, when we compute the probability of the American League team winning 4 games in a row, we find that it is also 0.0625. Therefore, probability that the series ends in four games would be 0.0625 + 0.0625 = 0.125; since the series would end if either the American or National League team won 4 games in a row.

Now let's tackle the question of finding probability that the world series ends in 5 games. The trick in finding this solution is to recognize that the series can only end in 5 games, if one team has won 3 out of the first 4 games. So let's first find the probability that the American League team wins exactly 3 of the first 4 games.

b(3; 4, 0.5) = 4C3 * (0.5)3 * (0.5)1 = 0.25

Okay, here comes some more tricky stuff, so listen up. Given that the American League team has won 3 of the first 4 games, the American League team has a 50/50 chance of winning the fifth game to end the series. Therefore, the probability of the American League team winning the series in 5 games is 0.25 * 0.50 = 0.125. Since the National League team could also win the series in 5 games, the probability that the series ends in 5 games would be 0.125 + 0.125 = 0.25.

The rest of the problem would be solved in the same way. You should find that the probability of the series ending in 6 games is 0.3125; and the probability of the series ending in 7 games is also 0.3125.

If you would like to cite this web page, you can use the following text:

Berman H.B., "Binomial Probability Distribution", [online] Available at: https://stattrek.com/probability-distributions/binomial URL [Accessed Date: 11/13/2022].