The local diner employs three servers, Ally, Benny, and Charli. On Monday night Ally served 40% of the customers, while Benny served 27%. What percent did Charli serve?

Question

a. The local diner employs three servers, Ally, Benny, and Charli. On Monday night Ally served 40% of the customers, while Benny served 27%. What percent did Charli serve?

b. That night 60% of Ally's customers ordered pie, while 40% did not. Of Benny's customers, 20% ordered pie. What percent did not?

c. Find the probability that a customer ordered pie on Monday night.

d. Given that a customer ordered pie that night, what is the probability that their server was Ally? Round to the nearest whole percent.

Chart: https://docs.google.com/document/d/19YmNm_F0QQjNLbhAs2Qzr8wI_hYZ78bdjp4IfPymoHo/edit?usp=sharing

Answer 1

Part A. Probabilities of being served by a particular waiter.

P(Ally) = .60 P(Benny) = 0.27 P(Charli) = 1 - P(Ally) - P(Benny) or 0.33 (33%)

Part B: We want to find the probability of one occurrence given a second occurance (a condition). We are given that the probability of ordering pie given that Ally was the server was 60% [which we write as P(Pie|Ally) = .60]. NOT P(Pie|Ally) = (1 - P(Pie|Ally)) which is 0.40

The calculations for Benny are the same. Probability of NOT getting pie if Benny is the server = 1 - 0.2 or 0.8

For the next part of the problem we MUST use the given chart to see what percentage of Charlie's customers ordered pie. (answer = .3)

Also, remember we have the following already.

P(A) = .4 P(B) = .27 P(C) = .33 P ( Pi | A ) = .6 P (Pi | B) = .2 P (Pi | C) = .3

We are looking for the conditional probability for Ally being a server given Pie was ordered. {P (Ally | Pie) }

We use Bayes's formula P(A | Pi) = ( P( Pi | A) * P ( A ) ) / P (Pi) Where Pi is pie and A is Ally

P( Pi | A) = .6 ; P ( A ) = .4 ; P (Pi) must still be calculated.

Probability of pie being ordered = P(Pi | Ally) * P(Ally) + P(Pi | Benny) * P(Benny) =

.4*.6 + .27 * .2 + .33 * .30 = .393

NOW we can answer the question.

Part C Probability that Ally was the server given that pie was ordered [ P(Ally | Pie) ]

P (A | Pi) = ( P( Pi | A) * P ( A ) ) / P (Pi) = .6 * .4 / .393 = 0.61069 or 61.1%