Question

Harrison Water Sports has two retail outlets: Seattle and Portland. The Seattle store does 60 percent of the total sales in a year. Further analysis indicates that if a sale is made in Seattle, there is 0.40 probability this is a sale of boat accessories, while if a sale is made at the Portland store this probability is 0.20. If Harrison Water Sports knows that a boat accessory has been sold, what is the probability this sale has occurred in Seattle:

Answer 1

Answer: P = 0.75

Explanation:

Hi!

The sample space of this problems is the set of all the possible sales. It is divided in the disjoint sets:

`S_s = {\text{sales made in Seattle }}\\S_p={\text {sales made in Portland}}`

We have also the set of sales of boat accesories
`S_b`, the colored one in the image.

We are given the data:

`P(S_s) = 0.6\\P(S_b | S_s) = (P(S_b\bigcap S_s))/(P(S_s))=0.4\\P(S_b|S_p) =(P(S_b\bigcap S_p))/(P(S_p))=0.2`

From these relations you can compute the probabilities of the intersections colored in the image:

`pink\;set:\;P(S_b \bigcap S_s) =0.6*0.4=0.24\\blue\;set\;:P(S_b \bigcap S_p)=(1-0.6)*0.2 =0.08`

You are asked about the conditional probability:

`P(S_s|S_b) = (P(S_s \bigcap S_b))/(P(S_b))`

To calculate this, you need
`P(S_b)` . In the image you can see that the set
`S_b` is the union of the two disjoint pink and blue sets. Then:

`P(S_b)=P((S_b \bigcap S_s)\bigcup(S_b \bigcap S_p)) = 0.24 + 0.08 = 0.32`

Finally:

`P(S_s|S_b) = (0.24)/(0.32)=0.75`