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]$n=6, x=3, p=\frac{1}{6}$[/tex]
Answer (2)
Use the binomial probability formula: P ( x ) = ( x n ) p x ( 1 − p ) ( n − x ) .
Calculate the binomial coefficient: ( 3 6 ) = 20 .
Calculate p x and ( 1 − p ) ( n − x ) : ( 6 1 ) 3 = 216 1 and ( 6 5 ) 3 = 216 125 .
Substitute the values into the formula and round to three decimal places: P ( 3 ) = 20 × 216 1 × 216 125 ≈ 0.054 .
Explanation
Understand the problem We are given a binomial distribution with n = 6 trials, x = 3 successes, and a probability of success on a single trial p = 6 1 . We want to find the probability of getting exactly 3 successes in 6 trials.
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.
Calculate the binomial coefficient First, we calculate the binomial coefficient: ( 3 6 ) = 3 ! ( 6 − 3 )! 6 ! = 3 ! 3 ! 6 ! = ( 3 × 2 × 1 ) ( 3 × 2 × 1 ) 6 × 5 × 4 × 3 × 2 × 1 = 3 × 2 × 1 6 × 5 × 4 = 20
Calculate p^x Next, we calculate p x :
p x = ( 6 1 ) 3 = 6 3 1 = 216 1
Calculate (1-p)^(n-x) Then, we calculate ( 1 − p ) n − x :
( 1 − p ) n − x = ( 1 − 6 1 ) 6 − 3 = ( 6 5 ) 3 = 6 3 5 3 = 216 125
Calculate the probability Now, we plug these values into the binomial probability formula: P ( 3 ) = ( 3 6 ) p 3 ( 1 − p ) 6 − 3 = 20 × 216 1 × 216 125 = 216 × 216 20 × 125 = 46656 2500 Now, we simplify the fraction: 46656 2500 = 11664 625 ≈ 0.05358
Round to three decimal places Finally, we round the result to three decimal places: 0.05358 ≈ 0.054
State the final answer Therefore, the probability of getting exactly 3 successes in 6 trials is approximately 0.054.
Examples
Consider a quality control process where 6 items are sampled from a production line, and the probability that any one item is defective is 6 1 . The binomial probability formula helps us calculate the likelihood of finding exactly 3 defective items in the sample. This is crucial for maintaining quality standards and making informed decisions about the production process. For instance, if the probability of finding 3 defective items is unacceptably high, the manufacturing process might need adjustments to reduce the defect rate. This kind of analysis is essential in many industries to ensure product reliability and customer satisfaction.
Using the binomial probability formula, the probability of achieving exactly 3 successes in 6 trials with a single trial success probability of 6 1 is calculated to be approximately 0.054 after rounding to three decimal places.
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