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## Binomial Probability Distribution

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

## Binomial Experiment

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

- The experiment consists of
*n*repeated trials. - Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
- The probability of success, denoted by
*P*, is the same on every trial. - 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 experiment consists of repeated trials. We flip a coin 2 times.
- Each trial can result in just two possible outcomes – heads or tails.
- The probability of success is constant – 0.5 on every trial.
- 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.

*x*: The number of successes that result from the binomial experiment.*n*: The number of trials in the binomial experiment.*P*: The probability of success on an individual trial.*Q*: The probability of failure on an individual trial. (This is equal to 1 –*P*.)*n!*: The factorial of n (also known as n factorial).- b(
*x*;*n, P*): Binomial probability – the probability that an*n*-trial binomial experiment results in exactly*x*successes, when the probability of success on an individual trial is*P*. _{n}C_{r}: The number of combinations of*n*things, taken*r*at a time.

## Binomial Distribution

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.

Number of heads | Probability |
---|---|

0 | 0.25 |

1 | 0.50 |

2 | 0.25 |

The binomial distribution has the following properties:

- The mean of the distribution (μ
_{x}) is equal to*n***P*. - The variance (σ
^{2}_{x}) is*n***P** ( 1 –*P*). - The standard deviation (σ
_{x}) is sqrt[*n***P** ( 1 –*P*) ].

## 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*) = _{n}C_{x} * P^{x} * (1 – P)^{n – x}

or

b(*x*; *n, P*) = { n! / [ x! (n – x)! ] } * P^{x} * (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) = _{5}C_{2} * (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).

For example, we might be interested in the cumulative binomial probability of obtaining 45 or fewer heads in 100 tosses of a coin (see Example 1 below). This would be the sum of all these individual binomial probabilities.

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)

**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 probabilities, using the binomial formula. The sum of all these 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 = 45; 100, 0.5)

b(x < 45; 100, 0.5) = 0.184

**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:* This is a very tricky application of the binomial distribution. 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) = _{4}C_{4} * (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) = _{4}C_{3} * (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.

## Frequently-Asked Questions

### 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 a particular outcome will occur on any given trial 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.

Outcome, x | Binomial probability, P(X = x) | Cumulative probability, P(X < x) |
---|---|---|

0 Heads | 0.125 | 0.125 |

1 Head | 0.375 | 0.500 |

2 Heads | 0.375 | 0.875 |

3 Heads | 0.125 | 1.000 |

### What is the number of trials?

The number of trials refers to the number of attempts in a binomial experiment. The number of trials is equal to the number of successes plus the number of failures.

Suppose that we conduct the following binomial experiment. We flip a coin and count the number of Heads. In this experiment, Heads would be classified 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.

### 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.

### 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 Problems

- Suppose you toss a fair coin 12 times. What is the probability of getting exactly 7 Heads.
**Solution:**

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

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.)

- Suppose the probability that a college freshman will graduate is 0.6. Three college freshmen are randomly selected. What is the probability that at most two of these students will graduate?
**Solution:**

We know the following:- The number of trials is 3 (because we have 3 students).
- The number of successes is 2.
- The probability of success for any individual student is 0.6.

Therefore, we plug those numbers into the Binomial Calculator and hit the Calculate button. The calculator reports that the cumulative binomial probability is 0.784. That is the probability that two or fewer of these three students will graduate is 0.784. (Note that the calculator also displays the binomial probability – the probability that EXACTLY two of these students graduate. The binomial probability is 0.432.)

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