Question

Of all registered cars in Arizona, 21% violate the state emissions standard. Eleven cars are selected at random to take an emissions test.  

What is the probability exactly 5 of them violate the standard? Report your answer to 4 decimal places.

Answer 1

Solution:

Given,

p = 21% = 0.21

1 - p = 1 - 0.21 = 0.79

n = 11

X follows the Binomial(11 , 0.21)

Using binomial probability formula ,

P(X = x) = (_n C _x) * p^x * (1 - p)^{n - x} ; x = 0 ,1 , 2 , ....., n

P(Exactly 5) = P(X = 5)

=  (₁₁ C ₅) * 0.21⁵ * (0.79)^{11 - 5}

= {11!/[5!*(11- 5)!] } * 0.21⁵ * 0.79⁶

= {11! / 5! * 6! }  * 0.21⁵ * 0.79⁶

= 462 * 0.21⁵ * 0.79⁶

= 0.0459

The probability exactly 5 of them violate the standard is 0.0459