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]$n=10, x=2, p=\frac{1}{3}$[/tex]
Answer (1)
Use the binomial probability formula: P ( X = x ) = ( x n ) p x ( 1 − p ) ( n − x ) .
Calculate the binomial coefficient: ( 2 10 ) = 45 .
Calculate p x and ( 1 − p ) ( n − x ) : ( 3 1 ) 2 = 9 1 and ( 3 2 ) 8 = 6561 256 .
Substitute the values into the formula and compute the probability: P ( X = 2 ) = 45 × 9 1 × 6561 256 ≈ 0.195 .
Explanation
Understand the problem We are given a binomial distribution with n = 10 trials, x = 2 successes, and a probability of success p = 3 1 on a single trial. We want to find the probability of exactly 2 successes in 10 trials.
State the binomial probability formula The binomial probability formula is given by: P ( X = x ) = ( x n ) p x ( 1 − p ) n − x where ( x n ) = x ! ( n − x )! n ! is the binomial coefficient.
Calculate the binomial coefficient First, we calculate the binomial coefficient: ( 2 10 ) = 2 ! ( 10 − 2 )! 10 ! = 2 ! 8 ! 10 ! = 2 × 1 10 × 9 = 45
Calculate p^x Next, we calculate p x :
p x = ( 3 1 ) 2 = 9 1
Calculate (1-p)^(n-x) Then, we calculate ( 1 − p ) n − x :
( 1 − p ) n − x = ( 1 − 3 1 ) 10 − 2 = ( 3 2 ) 8 = 3 8 2 8 = 6561 256
Calculate the final probability Now, we plug these values into the binomial probability formula: P ( X = 2 ) = 45 × 9 1 × 6561 256 = 5 × 6561 256 = 6561 1280 ≈ 0.195
State the final answer Therefore, the probability of getting exactly 2 successes in 10 trials is approximately 0.195.
Examples
Consider a basketball player who makes a free throw with a probability of 3 1 . If the player attempts 10 free throws, the probability of making exactly 2 of them can be calculated using the binomial probability formula. This type of problem is useful in sports analytics to understand the likelihood of certain outcomes based on individual player statistics. Understanding binomial distributions can help coaches and players make strategic decisions based on probabilities.
