Question

Can someone please explain detail steps to solve this problem? I need a better understanding.

1. A fast-food restaurant manager knows that cars arrive at the drive-through at the rate of 3 cars per minute between the hours of 12 noon and 12.30 pm. The manager wants to determine and interpret the probability of following events; a) Exactly 6 cars arrive between 12 noon and 12.05 pm. b) Fewer than 6 cars arrive between 12.00 noon and 12.05 pm. c) At least 6 cars arrive between 12 noon and 12.05 pm.

Answer 1

Given that  cars arrive at the drive-through at the rate of 3 cars per minute between the hours of 12 noon and 12.30 pm

Average no of cars that arrive between 12.00 noon and 12.05 pm = 5*3 = 15

Let us assume that distribution of no of cars that arrive between 12.00 noon and 12.05 pm follow Poisson distribution.

λ = 15

a)

Probability that exactly 6 cars arrive between 12.00 noon and 12.05 pm

b)

Probability that fewer than 6 cars arrive between 12.00 noon and 12.05 pm = P(X < 6)

= P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)

= 0.00279

c) Probability that at least 6 cars arrive between 12 noon and 12.05 pm.

= 1 - P(X < 6)

= 1 - 0.00279 = 0.99721