Question

PLEASE SHOW WORK

According to a​ newspaper, 50​% of bicycles stolen in a certain country are recovered. Find the probability​ that, in a sample of 66 randomly selected cases of bicycles stolen in the​ country, exactly 22 out of 66 bikes are recovered. Explain what you are using to calculate this and verify that all the conditions are met to use it. ​ Then, describe your solution in terms of the problem.

Answer 1

X ~ Binomial (n,p)

Where n = 66 , p = 0.50

np = 66 * 0.50 = 33 >= 5 , n(1-p) = 66 * 0.5 = 33 >= 5

Since np and n(1-p) both greater than 5, normal approximation is suitable.

Using normal approximation to binomial distribution ,

P( X < x) = P( Z < x - np / sqrt(np(1-p) ) )

Using continuity correction for normal approximation,

P( X = x) = P( x - 0.5 < X < x + 0.5)

Therefore,

P( X = 22) = P(21.5 < X < 22.5)

= P( X < 22.5) - P( X < 21.5)

= P( Z < 22.5 - 66 * 0.5 / sqrt( 66 * 0.5 * 0.5) ) - P( Z < 21.5 - 66 * 0.5 / sqrt( 66 * 0.5 * 0.5) )

= P( Z < -2.5849) - P( Z < -2.8311)

= ( 1 - P( Z < 2.5849) ) - ( 1 - P( Z < 2.8311) )

= ( 1 - 0.9951 ) - ( 1 - 0.9977)

= 0.0026

Probability that exactly 22 out of 66 bikes are recovered = 0.0026