Question

A researcher wishes to conduct a study of the color preferences of new car buyers. Suppose that 50% of this population prefers the color red. If 20 buyers are randomly selected, what is the probability that exactly 15 buyers would prefer red? Round your answer to four decimal places.

Answer 1

Probability that exactly 15 buyers out of 20 will prefer red is 0.0148.

Explanation:

This problem has binomial probability distribution.

So probability of success ( population preferring Red colour) i.e. p = 50% = 0.5

therefore probability of failure ( population preferring other colors ) i.e. q = 1 - p = 1 - 0.5 = 0.5

20 buyers are selected at random, therefore n = 20.

We need to find the probability that exactly 15 buyers would prefer red and so r = 15 and it is given by,

P( X = r ) =
`n_C_r * p^(r)* q^(n-r)`

So, P ( X = 15 ) =
`{20}_C_(15) * 0.5^(15)* 0.5^(20-15)` =
`{20}_C_(15) * 0.5^(15)* 0.5^(5)` =
`0.0148`