Question

4. Historically, 45% of visitors to the online store of XYZ company made a purchase.

40% of the visitors who made a purchase came to the online store through a web search engine. Also, 15% of visitors who did not make a purchase came to the online store through a web search engine. If a visitor comes to the online store through a web search engine, what is the probability that he or she will make a purchase? (Suggestion: To think through the problem, use a probability tree and “whether a visitor makes a purchase” as the first step to depict what is given. Use the Bayes Theorem to calculate the probability in question. You do not need to include the tree in your submission.). (4 points)

Answer 1

We are given here that:
P( purchase ) = 0.45, Therefore P(do not purchase ) = 1 - 0.45 = 0.55

Also, we are given the conditional probabilities here:
P( web search engine | purchase) = 0.4,
P( web search engine | do not purchase) = 0.15

Therefore, using law of addition of probability, we have here:
P( web search engine) = P( web search engine | purchase) P(purchase) + P( web search engine | do not purchase)P(do not purchase)

= 0.4*0.45 + 0.15*0.55

= 0.2625

Now given that a user can through a web search engine, probability that the user will make a purchase is computed using Bayes theorem here as:

P( purchase | web search engine) = P( web search engine | purchase) P(purchase) / P( web search engine)

= 0.4*0.45 / 0.2625

= 0.6857

Therefore 0.6857 is the required probability here.