### Introduction

There are three characteristics of a **binomial experiment**:

- There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter
*n*denotes the number of trials. - There are only two possible outcomes, called
*success*and*failure*, for each trial. The outcome that we are measuring is defined as a*success*, while the other outcome is defined as a*failure*. The letter*p*denotes the probability of a success on one trial, and*q*denotes the probability of a failure on one trial.*p*+*q*= 1. - The
*n*trials are independent and are repeated using identical conditions. Because the*n*trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability,*p*, of a success and probability,*q*, of a failure remain the same. Let us look at several examples of a binomial experiment.*n*= 1. There are only two outcomes, a head or a tail, of each trial. We can define a head as a success if we are measuring number of heads. For a fair coin, the probabilities of getting a head or a tail are both .5. So,*p*=*q*− .5. Both*p*and*q*remain the same from trial to trial. This experiment is also called a**Bernoulli trial**, named after Jacob Bernoulli who, in the late 1600s, studied such trials extensively. Any experiment that has characteristics two and three and where*n*= 1 is called a**Bernoulli trial**. A binomial experiment takes place when the number of successes is counted in one or more Bernoulli trials.*n*= 1. There are only two outcomes, guess correctly or guess wrong, of each trial. We can define guess correctly as a success. For a random guess (you have no clue at all), the probability of guessing correct should be $\frac{1}{4}$ because there are four options and only one option is correct. So, and $q=\mathrm{}1-p=\mathrm{}1-\frac{1}{4}=\frac{3}{4}$. Both*p*and*q*remain the same from trial to trial. This experiment is also a Bernoulli trial. It meets the characteristics two and three and*n*= 1.*n*= 5. There are only two outcomes, a head or a tail, of each trial. If we define a head as a success, then*p*=*q*= 0.5. Both*p*and*q*remain the same for each trial. Since*n*= 5, this experiment is not a Bernoulli trial although it meets the characteristics two and three.*n*= 10. There are only two outcomes, guess correctly or guess wrong, of each trial. We can define guess correctly as a success. As we explained in example 2, $p=\frac{1}{4}$ and $q=\mathrm{}1-p=\mathrm{}1-\frac{1}{4}=\frac{3}{4}$. Both*p*and*q*remain the same for each guess. Since*n*= 10, this experiment is not a Bernoulli trial.*n*= 2. There are only two outcomes, a red ball or a blue ball, of each trial. If we define selecting a red ball as a success, then selecting a blue ball is a failure. The probability of getting the first ball red is $\frac{5}{10}$ since there are five red balls out of 10 balls. So, $p=\frac{5}{10}$ and $q=\mathrm{}1-p=\mathrm{}1-\frac{5}{10}=\frac{5}{10}$. However,*p*and*q*do not remain the same for the second trial. If the first ball selected is red, then the probability of getting the second ball red is $\frac{4}{9}$ since there are only four red balls out of nine balls. But if the first ball selected is blue, then the probability of getting the second ball red is $\frac{5}{9}$ since there are still five red balls out of nine balls.*n*is not fixed.*n*could be 1 if a head appears from the first toss.*n*could be 2 if the first toss is a tail and the second toss is a head. So on and so forth.

The outcomes of a binomial experiment fit a binomial probability distribution. The random variable *X* = the number of successes obtained in the *n* independent trials.

There are shortcut formulas for calculating mean *μ*, variance *σ*^{2}, and standard deviation *σ* of a binomial probability distribution. The formulas are given as below. The deriving of these formulas will not be discussed in this book.

Here *n* is the number of trials, *p* is the probability of a success, and *q* is the probability of a failure.

### Example 4.8

At ABC High School, the withdrawal rate from an elementary physics course is 30 percent for any given term. This implies that, for any given term, 70 percent of the students stay in the class for the entire term. The random variable *X* = the number of students who withdraw from the randomly selected elementary physics class. Since we are measuring the number of students who withdrew, a *success* is defined as an individual who withdrew.

The state health board is concerned about the amount of fruit available in school lunches. Forty-eight percent of schools in the state offer fruit in their lunches every day. This implies that 52 percent do not. What would a *success* be in this case?

### Example 4.9

Suppose you play a game that you can only either win or lose. The probability that you win any game is 55 percent, and the probability that you lose is 45 percent. Each game you play is independent. If you play the game 20 times, write the function that describes the probability that you win 15 of the 20 times. Here, if you define *X* as the number of wins, then *X* takes on the values 0, 1, 2, 3, . . ., 20. The probability of a success is *p* = 0.55. The probability of a failure is *q* = .45. The number of trials is *n* = 20. The probability question can be stated mathematically as *P*(*x* = 15). If you define *X* as the number of losses, then a *success* is defined as a loss and a *failure* is defined as a win. A *success* does not necessarily represent a good outcome. It is simply the outcome that you are measuring. *X* still takes on the values of 0, 1, 2, 3, . . ., 20. The probability of a success is $p=.45$. The probability of a failure is $q=.55$.

A trainer is teaching a dolphin to do tricks. The probability that the dolphin successfully performs the trick is 35 percent, and the probability that the dolphin does not successfully perform the trick is 65 percent. Out of 20 attempts, you want to find the probability that the dolphin succeeds 12 times. State the probability question mathematically.

### Example 4.10

A fair coin is flipped 15 times. Each flip is independent. What is the probability of getting more than 10 heads? Let *X* = the number of heads in 15 flips of the fair coin. *X* takes on the values 0, 1, 2, 3, ..., 15. Since the coin is fair, *p* = .5 and *q* = .5. The number of trials *n* = 15. State the probability question mathematically.

*P*(*x* > 10)

A fair, six-sided die is rolled 10 times. Each roll is independent. You want to find the probability of rolling a one more than three times. State the probability question mathematically.

### Example 4.11

Approximately 70 percent of statistics students do their homework in time for it to be collected and graded. Each student does homework independently. In a statistics class of 50 students, what is the probability that at least 40 will do their homework on time? Students are selected randomly.

a. This is a binomial problem because there is only a success or a ________, there are a fixed number of trials, and the probability of a success is .70 for each trial.

a. failure

b. If we are interested in the number of students who do their homework on time, then how do we define *X*?

b. *X* = the number of statistics students who do their homework on time

c. What values does *x* take on?

c. 0, 1, 2, . . ., 50

d. What is a *failure*, in words?

d. Failure is defined as a student who does not complete his or her homework on time.

The probability of a success is *p* = .70. The number of trials is *n* = 50.

e. If *p* + *q* = 1, then what is *q*?

e. *q* = .30

f. The words *at least* translate as what kind of inequality for the probability question *P*(*x* ____ 40)?

f. greater than or equal to (≥)

The probability question is*P*(

*x*≥ 40).

Sixty-five percent of people pass the state driver’s exam on the first try. A group of 50 individuals who have taken the driver’s exam is randomly selected. Give two reasons why this is a binomial problem.