# 5.3 Binomial distribution - university of calgary - base content  (Page 2/30)

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A fair coin is flipped 15 times. Each flip is independent. What is the probability of getting more than ten 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 = 0.5 and q = 0.5. The number of trials is n = 15. State the probability question mathematically.

P ( X >10)

## Try it

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

P ( X >3)

Approximately 70% 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 0.70 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 = 0.70. The number of trials is n = 50.

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

e. q = 0.30

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).

## Try it

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.

This is a binomial problem because there is only a success or a failure, and there are a definite number of trials. The probability of a success stays the same for each trial.

## Notation for the binomial: B = binomial probability distribution function

X ~ B ( n , p )

Read this as " X is a random variable with a binomial distribution." The parameters are n and p ; n = number of trials, p = probability of a success on each trial.

## Binomial formula

P ( X = x )= $\left(\begin{array}{c}n\\ x\end{array}\right)p^{x}q^{\mathrm{n-x}}$

• $p^{x}$ is the probability of x successes in x independent and identical trials. For example, if the probability of success = 0.4, the probability of five successes in five independent and identical trials is
$0.4\times 0.4\times 0.4\times 0.4\times 0.4=\mathrm{0.4}^{5}=p^{x}$
• $q^{\mathrm{n-x}}$ is the probability of n - x failures in n - x identical and independent trials. For example, if the probability of success = 0.4, then the probability of failure is 1 - p = 0.6. If there are eight trials ( n = 8) with five successes ( x = 5 ), then there were three failures in the eight trials ( n - x = 8 - 5 = 3). The probability of three failures in three independent and identical trials is
$0.6\times 0.6\times 0.6=\mathrm{0.6}^{3}=q^{\mathrm{n-x}}$
• $\left(\begin{array}{c}n\\ x\end{array}\right)$ represents the number of combinations of x successes in n trials. If there are eight trials ( n = 8) and five successes ( x = 5), then there are 56 possible ways to arrange five successes among eight trials.
$\left(\begin{array}{c}8\\ 5\end{array}\right)=\mathrm{56}$
• The formula P ( X = x )= $\left(\begin{array}{c}n\\ x\end{array}\right)p^{x}q^{\mathrm{n-x}}$ is the probability of x successes in n independent and identical trials. If there are eight independent and identical trials, the probability of five successes where p = 0.4 is
$P(\mathrm{X=5})=\left(\begin{array}{c}8\\ 5\end{array}\right)\mathrm{0.4}^{5}\mathrm{0.6}^{3}=\mathrm{0.1239}$

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