A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.
[tex]$n=10, p=0.45, x=8$[/tex]
[tex]$P(8)= \square$[/tex]
(Do not round until the final answer. Then round to four decimal places as needed.)
Answer (1)
Calculate the binomial coefficient: ( 8 10 ) = 45 .
Calculate p x = ( 0.45 ) 8 = 0.00016814672 .
Calculate ( 1 − p ) ( n − x ) = ( 0.55 ) 2 = 0.3025 .
Calculate the probability: P ( 8 ) = 45 × 0.00016814672 × 0.3025 ≈ 0.0023 .
Explanation
Problem Setup We are given a binomial probability experiment with n = 10 trials, a success probability of p = 0.45 , and we want to find the probability of x = 8 successes. We will use the binomial probability formula to calculate this.
Binomial Probability Formula The binomial probability formula is given by: P ( x ) = ( x n ) p x ( 1 − p ) n − x where ( x n ) = x ! ( n − x )! n ! is the binomial coefficient.
Calculate Binomial Coefficient First, we calculate the binomial coefficient: ( 8 10 ) = 8 ! ( 10 − 8 )! 10 ! = 8 ! 2 ! 10 ! = 2 × 1 10 × 9 = 45
Calculate p^x Next, we calculate p x :
p x = ( 0.45 ) 8 = 0.00016814672
Calculate (1-p)^(n-x) Then, we calculate ( 1 − p ) n − x :
( 1 − p ) n − x = ( 1 − 0.45 ) 10 − 8 = ( 0.55 ) 2 = 0.3025
Calculate P(8) Now, we plug these values into the binomial probability formula: P ( 8 ) = 45 × ( 0.45 ) 8 × ( 0.55 ) 2 = 45 × 0.00016814672 × 0.3025 = 0.00228895894
Round to Four Decimal Places Finally, we round the result to four decimal places: P ( 8 ) ≈ 0.0023
Final Answer Therefore, the probability of 8 successes in 10 trials is approximately 0.0023.
Examples
Consider a basketball player who makes 45% of their shots. If they take 10 shots, the probability of them making exactly 8 of those shots can be calculated using the binomial probability formula. This type of calculation is useful in sports analytics for predicting player performance and understanding the likelihood of specific outcomes.
