Aaron, Bob, and Carl are the proud owners of Three Brothers Motorcycles. During

Question

Aaron, Bob, and Carl are the proud owners of Three Brothers Motorcycles. During business hours, customers arrive at a rate of 0.6 customers per hour. Business hours are from 9am until 6pm, with a lunch hour from noon until 1pm when the shop is closed. The brothers love to make bets among themselves. Upon opening one morning, Aaron says. "I bet we have exactly two customers come in before lunch". Bob replies, "I bet we have fewer than two customers before lunch". Carl chimes in, "I bet you guys are both wrong"! At this point the three brothers point at each other and exclaim in unison: "You're on! We have a bet". Let X be the number of customers that arrive before lunchtime. a) Who has the best chance of winning? ____ P(Aaron wins the bet) = P(X =___) =____ P(Bob wins the bet) = P(X

Explanation / Answer

Mean = 0.6 customers per hour
Time before lunch time = 3 hrs

Average number of customers arrival for 3 hrs = 0.6 * 3 = 1.8

Assuming the arrival rate of customers follow poisson distribution. P(X=x) = (e-) (x) / x!

P(Aaron wins the bet) = P( X = 2) = exp(-1.8) (1.8^2) / 2! = 0.2678
P(Bob wins the bet) = P(X < 2) = P(X=0) + P(X = 1) = exp(-1.8) + 1.8exp(-1.8) = 0.4628
P(Carl wins the bet) = P(X > 2) = 1 - P( X = 2) - P(X > 2) = 1 - 0.2678 - 0.4628 = 0.2694

So, Bob wins the bet.

(b) Assuming that time between arrivals follow exponential distribution with mean = 0.6 customers per hour

Time before 2 pm is 1 hr.

Time between 2 pm and 3.45 pm from 1 pm is between 1 hr to 1.75 hr

P(Aaron wins the bet) = P( Y < 1) = 1 - exp(-0.6*1) = 0.4512
P(Bob wins the bet) = P(1 < Y < 1.75) = P(Y < 1) - P( Y < 1.75) = (1 - exp(-0.6))  - (1 - exp(-0.6*1.75))

= exp(-0.6) - exp(-1.05) = 0.1989
P(Carl wins the bet) = 1 - P( Y < 1) - P(1 < Y < 1.75) = 1 - 0.4512 - 0.1989 = 0.35

So, Aaron wins the bet.

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