To maximize the probability that customers purchase products during a visit to a website, a company that provides marketing services and website management experiments with different offers and website design to understand customer Web behavior. Possible actions by the website include offering the customer an instant discount, offering the customer free shipping, or doing nothing. A recent experiment found that customers make purchases 6% of the time when offered the instant discount, 5% when offered free shipping, and 3% when no special offer was given. Suppose 30% of the customers are offered the discount and an additional 40% are offered free shipping. A) Construct a probability tree for this experiment. B) What recent of customers who visit the site made purchases? C) Given that a customer made a purchase, what is the probability that they were offered free shipping?

A. In order to understand the probability tree, we have to define three different scenarios that could happen when certain customer visits the website:

1. Discount (30%)

2. Free shipping (40%)

3. No special offer, (it must be 30% because all gives a total of 100%)

Then, for each scenario two things could happen, customer makes a purchase or not. Given the information above, each branch of the tree has the specified percentage of occurrence or probability.

B. Once the probability tree is made, the next step is to multiply the percentages of each consecutive branch. For example:

The percentage of customers that visit the website, receive free shipping and make a purchase is 40% x 5 %, as it has been specified in the picture.

Then, after calculate all percentages of customers that make a purchase, it is necessary to add them to finally get a total of 4.7%.

C. We can understand it as follows:

The customers are divided between three main sets:

Those who were offered with special discount.

Those who were offered with free shipping.

Those who received no special offer.

Inside these sets there is the set of costumers who made a purchase, and we have already that percentage, it is 4.7%. However, to determine the asked probability, we have to keep our attention in the intersection between the set of those who were offered with free shipping and those who made a purchase. The percentage of that tiny set over the percentage of customers who made a purchase, is the probability that we must determine:

P ( B│A)=(P(A∩B))/(P(A))=0.02/0.047=0.42

Where A: the costumer made a purchase

B: the costumer received an offer of free shipping

The probability that a costumer were offered with free shipping given that he made a purchase is 0.42.