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: P = 0.75Step-by-step explanation:Hi!The sample space of this problems is the set of all the possible sales. It is divided in the disjoint sets:[tex]S_s = {\text{sales made in Seattle }}\\S_p={\text {sales made in Portland}}[/tex]We have also the set of sales of boat accesories [tex]S_b[/tex], the colored one in the image.We are given the data:[tex]P(S_s) = 0.6\\P(S_b | S_s) = \frac{P(S_b\bigcap S_s)}{P(S_s)}=0.4\\P(S_b|S_p) =\frac{P(S_b\bigcap S_p)}{P(S_p)}=0.2[/tex]From these relations you can compute the probabilities of the intersections colored in the image:[tex]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[/tex]You are asked about the conditional probability:[tex]P(S_s|S_b) = \frac{P(S_s \bigcap S_b)}{P(S_b)}[/tex]To calculate this, you need  [tex]P(S_b)[/tex] . In the image you can see that the set [tex]S_b[/tex] is the union of the two disjoint pink and blue sets. Then:[tex]P(S_b)=P((S_b \bigcap S_s)\bigcup(S_b \bigcap S_p)) = 0.24 + 0.08 = 0.32[/tex]Finally:[tex]P(S_s|S_b) = \frac{0.24}{0.32}=0.75[/tex]