Question

At Sally's Hair Salon there are three hair stylists. 36% of the hair cuts are done by Chris, 26% are done by Karine, and the rest are done by Amy. Chris finds that when he does hair cuts, 5% of the customers are not satisfied. Karine finds that when she does hair cuts, 6% of the customers are not satisfied. Amy finds that when she does hair cuts, 8% of the customers are not satisfied. Suppose that a customer leaving the salon is selected at random. If the customer is not satisfied, what is the probability that their hair was done by Amy

Answer 1

Probability that their hair was done by Amy is 0.475.

Explanation:

We are given that Sally's Hair Salon there are three hair stylists. 36% of the hair cuts are done by Chris, 26% are done by Karine, and the rest are done by Amy.

Let the Probability of hair cutting done by Chris = P(C) = 0.36

Probability of hair cutting done by Karine = P(K) = 0.26

Probability of hair cutting done by Amy = P(A) = 0.368

Also, let NS = event that customer is not satisfied with his cutting

So, Probability that customers are not satisfied given that their hair cutting is done by Chris = P(NS/C) = 0.05

Probability that customers are not satisfied given that their hair cutting is done by Karine = P(NS/K) = 0.06

Probability that customers are not satisfied given that their hair cutting is done by Amy = P(NS/A) = 0.08

Now, a customer leaving the salon is selected at random. If the customer is not satisfied, the probability that their hair was done by Amy is given by = P(A/NS)

For finding the above probability we will use the concept of Bayes' Theorem;

SO, P(A/NS) =
`\frac{\text{P(A) * P(NS/A)}}{\text{P(C) * P(NS/C) + P(K) * P(NS/K) +P(A) * P(NS/A) }}`
`(P(A) * P(NS/A) )/(P(C) * P(NS/C) +P(K) * P(NS/K) +P(A) * P(NS/A) )`

=
`(0.38 * 0.08)/(0.36 * 0.05+0.26 * 0.06+0.38 * 0.08)`

=
`(0.0304)/(0.064)`

= 0.475

Hence, the probability that their hair was done by Amy is 0.475.