Question

The following table shows the ratings of customer satisfaction at 1200 restaurants based on the capacity of the restaurant. Excellent Fair Poor Total Seats fewer than 100 182 203 165 550 Seats 100 or more 180 311 159 650 Total 362 514 324 1200 a) Find the fraction for the probability that a randomly selected restaurant seats fewer than 100 people. Leave your answer as a fraction. No need to simplify. 182/362 b) Find the fraction for the probability that a randomly selected restaurant received a poor rating. Leave your answer as a fraction. No need to simplify. 165/1200 c) Find the fraction for the probability that a randomly selected restaurant received an excellent rating, given that the restaurant seats fewer than 100 people. Leave your answer as a fraction. No need to simplify. 182/1200 d) Find the fraction for the probability that a randomly selected restaurant seats fewer than 100 people and received an excellent rating. Leave your answer as a fraction. No need to simplify.

Answer 1

Explanation:

a) The number of people rated for restaurants that seat fewer than 100 people is 550. The total number of people rated is 1200. Therefore, the probability that a randomly selected restaurant seats fewer than 100 people is

550/1200

b) The number of poor ratings for both restaurants is 324. Therefore, the probability that a randomly selected restaurant received a poor rating is

324/1200

c) This is a conditional probability. If A represents the event that a randomly selected restaurant received an excellent rating and B represents the event that the restaurant seats fewer than 100 people,

P(A) = 362/1200

P(B) = 550/1200

P(A and B) = 362/1200 × 550/1200 =

199100/1440000

The conditional probability required is

P(A|B) = P(A and B)/P(B)

P(A|B) = (199100/1440000)/550/1200

P(A|B) = 362/1200

d) If A represents the event that a randomly selected restaurant seats fewer than 100 people and B represents the event that the restaurant received an excellent rating,

P(A) = 550/1200

P(B) = 362/1200

P(A and B) = 362/1200 × 550/1200 =

199100/1440000