A car wash offers both a regular wash and a deluxe wash for their customers. They...

A car wash offers both a regular wash and a deluxe wash for their customers. They track their customers over time and determine that a customer orders the deluxe wash with a probability of 0.3. They also offer air fresheners for their customer for a nominal fee and find that regular wash customers get an air freshener with a probability of 0.45, and deluxe wash customers get an air freshener with probability 0.8. Assume that the car wash only offers the regular and deluxe packages.

a.) What is the probability that a customer gets an air freshener, regardless of the type of wash they ordered?

b.) What is the probability a customer ordered the deluxe package, provided that you know they purchased an air freshener?

Solutions

Expert Solution

P[ deluxe wash ] = 0.3

P[ regular wash ] = 1 - P[ deluxe wash ]

P[ regular wash ] = 1 - 0.3

P[ regular wash ] = 0.7

P[ air freshener | deluxe wash ] = 0.8

P[ air freshener | regular wash ] = 0.45

a.) What is the probability that a customer gets an air freshener, regardless of the type of wash they ordered?

P[ air freshener ] = P[ air freshener | regular wash ]*P[ regular wash ] + P[ air freshener | deluxe wash ]*P[ deluxe wash ]

P[ air freshener ] = 0.45*0.7 + 0.8*0.3

P[ air freshener ] = 0.315 + 0.24

P[ air freshener ] = 0.555

b.) What is the probability a customer ordered the deluxe package, provided that you know they purchased an air freshener?

P[ deluxe wash | air freshener ] = P[ air freshener | deluxe wash ]*P[ deluxe wash ] / P[ air freshener ]

P[ deluxe wash | air freshener ] = 0.8*0.3 /0.555

P[ deluxe wash | air freshener ] = 0.24/0.555

P[ deluxe wash | air freshener ] = 0.432

orchestra answered 1 year ago

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