Assume that a procedure yields a binomial distribution with a trial repeated [tex]$n$[/tex] times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial. Round to three decimal places.
[tex]$n=5, x=2, p=0.70$[/tex]
Answer (1)
Calculate the binomial coefficient: ( 2 5 ) = 10 .
Calculate the probability of success raised to the power of the number of successes: ( 0.70 ) 2 = 0.49 .
Calculate the probability of failure raised to the power of the number of failures: ( 0.30 ) 3 = 0.027 .
Apply the binomial probability formula and round the result: P ( 2 ) = 10 × 0.49 × 0.027 ≈ 0.132 .
Explanation
Understand the problem and provided data We are given a binomial distribution problem where we need to find the probability of getting exactly x = 2 successes in n = 5 trials, with the probability of success on a single trial being p = 0.70 . We will use the binomial probability formula to solve this problem.
State the binomial probability formula The binomial probability formula is given by: P ( x ) = ( x n ) p x ( 1 − p ) n − x where:
P ( x ) is the probability of getting exactly x successes in n trials,
( x n ) is the binomial coefficient, which represents the number of ways to choose x successes from n trials,
p is the probability of success on a single trial,
( 1 − p ) is the probability of failure on a single trial,
n is the number of trials, and
x is the number of successes.
Calculate the binomial coefficient First, we need to calculate the binomial coefficient ( x n ) , which is the number of ways to choose x = 2 successes from n = 5 trials: ( 2 5 ) = 2 ! ( 5 − 2 )! 5 ! = 2 ! 3 ! 5 ! = ( 2 × 1 ) ( 3 × 2 × 1 ) 5 × 4 × 3 × 2 × 1 = 2 × 1 5 × 4 = 10 So, there are 10 ways to get 2 successes in 5 trials.
Calculate p x Next, we calculate p x , which is the probability of success raised to the power of the number of successes: p x = ( 0.70 ) 2 = 0.49
Calculate ( 1 − p ) ( n − x ) Now, we calculate ( 1 − p ) ( n − x ) , which is the probability of failure raised to the power of the number of failures: ( 1 − p ) ( n − x ) = ( 1 − 0.70 ) ( 5 − 2 ) = ( 0.30 ) 3 = 0.027
Calculate the final probability and round the result Now, we substitute all the calculated values into the binomial probability formula: P ( 2 ) = ( 2 5 ) p 2 ( 1 − p ) 5 − 2 = 10 × 0.49 × 0.027 = 0.1323 Finally, we round the result to three decimal places: P ( 2 ) ≈ 0.132
State the final answer Therefore, the probability of getting exactly 2 successes in 5 trials, with the probability of success on a single trial being 0.70, is approximately 0.132.
Examples
Consider a basketball player who makes 70% of their free throws. If they take 5 free throws, the probability of them making exactly 2 of those free throws is approximately 0.132. This calculation is useful for coaches and players to understand the likelihood of different outcomes during a game, helping them to strategize and manage expectations. Understanding binomial probabilities can also be applied to quality control in manufacturing, where you might want to know the probability of finding a certain number of defective items in a batch.
