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 [tex]$x$[/tex] successes given the probability [tex]$p$[/tex] of success on a single trial. Round to three decimal places.
[tex]$n=4, x=3, p=\frac{1}{6}$[/tex]
Answer (2)
Use the binomial probability formula: P ( x ) = ( x n ) p x ( 1 − p ) ( n − x ) .
Substitute n = 4 , x = 3 , and p = 6 1 into the formula.
Calculate P ( 3 ) = ( 3 4 ) ( 6 1 ) 3 ( 1 − 6 1 ) ( 4 − 3 ) = 4 ∗ 216 1 ∗ 6 5 = 324 5 .
Convert to decimal and round: 324 5 ≈ 0.015 .
Explanation
Understand the problem and provided data We are given a binomial distribution with n = 4 trials, and we want to find the probability of x = 3 successes, where the probability of success on a single trial is p = 6 1 . 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 ( x n ) = x ! ( n − x )! n ! is the binomial coefficient.
Plug in the values Now, we plug in the given values: n = 4 , x = 3 , and p = 6 1 into the formula: P ( 3 ) = ( 3 4 ) ( 6 1 ) 3 ( 1 − 6 1 ) 4 − 3
Calculate the binomial coefficient First, we calculate the binomial coefficient: ( 3 4 ) = 3 ! ( 4 − 3 )! 4 ! = 3 ! 1 ! 4 ! = ( 3 × 2 × 1 ) ( 1 ) 4 × 3 × 2 × 1 = 4
Calculate p^x Next, we calculate p x :
( 6 1 ) 3 = 6 3 1 = 216 1
Calculate (1-p)^(n-x) Then, we calculate ( 1 − p ) ( n − x ) :
( 1 − 6 1 ) 4 − 3 = ( 6 5 ) 1 = 6 5
Calculate P(3) Now, we plug these values back into the binomial probability formula: P ( 3 ) = 4 × 216 1 × 6 5 = 216 × 6 4 × 1 × 5 = 1296 20 = 324 5
Convert to decimal and round Finally, we convert the fraction to a decimal and round to three decimal places: 324 5 ≈ 0.015432 ≈ 0.015
State the final answer Therefore, the probability of getting exactly 3 successes in 4 trials, with the probability of success on a single trial being 6 1 , is approximately 0.015.
Examples
Consider a quality control process where 4 items are sampled from a production line, and the probability of an item being defective is 6 1 . We want to find the probability that exactly 3 of the 4 sampled items are defective. This is a direct application of the binomial probability formula. Knowing this probability helps in assessing the effectiveness of the quality control measures and making informed decisions about the production process. For example, if the probability of 3 defective items is unacceptably high, the production process may need adjustments.
Using the binomial probability formula, the probability of getting exactly 3 successes in 4 trials with a success probability of p = 6 1 is calculated to be approximately 0.015. This is derived through the steps of calculating the binomial coefficient, the probability of successes, and the failure probability, leading to the final result. Rounding the decimal gives us the answer for the probability.
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