Question

A brand name has a 40​% recognition rate. If the owner of the brand wants to verify that rate by beginning with a small sample of 5 randomly selected​ consumers, find the probability that exactly 2 of the 5 consumers recognize the brand name. Also find the probability that the number who recognize the brand name is not 2.

Answer 1

1) 0.3456

2) 0.6544.

Explanation:

Let X represents the event of recognizing the brand,

Given,

The probability of recognizing the brand, p = 40​% = 0.40,

Thus, the probability of not recognizing the brand, q = 1 - 0.40 = 0.60,

Since, the binomial distribution formula,

`P(x) = ^nC_r (p)^r(q)^(n-r)`

Where,

`^nC_r=(n!)/(r!(n-r)!)`

1) Thus, the probability that exactly 2 of the 5 consumers recognize the brand name is,

`P(X=2)=^5C_2 (0.40)^2 (0.60)^(5-2)`

`=10 (0.40)^2 (0.60)^3`

`=0.3456`

2) Also, the probability that the number who recognize the brand name is not 2 = 1 - P(X=2) = 1 - 0.3456 = 0.6544.