2. Mercy has purchased a BMW 760i. She puts its key into one of the 3empty boxes

Question

2. Mercy has purchased a BMW 760i. She puts its key into one of the 3empty boxes: box 1, box 2, and box 3. She has closed all the boxes, so you do not know which box has the key. She will give you the new car if you select the box that contains thekey.

You select box 1 but do not open it. Of the remaining 2 boxes, she opens the box that doesnot have the key. She opens box 3. She gives you the option of replacing your box 1 with box 2.

a. Calculate the probability that box 1 contains the key

.b. Calculate the probability that box 2 contains the key.

Several methods for calculating the above probabilities exist. One method uses a tree diagram with the following events.

     Ak = box k contains key

     Bk = Mercy opens box k

At the root of the tree is the event that you select box 1.

Explanation / Answer

Consider the events CK1, CK2 and CK3 indicating that the car key is behind respectively box 1, 2 or 3. These three events all have probability 1/3.

The player initially choosing box 1 is described by the event X1. As the first choice of the player is independent of the position of the car, also the conditional probabilities are P(Ci|X1) = 1/3. The host opening box 3 is described by H3. For this event it holds:

P(H3|CK1,X1) = ½;

P(H3|CK2,X1) = 1;

P(H3|CK3,X1) = 0

Then, if the player initially selects box 1, and the host opens box 3, the conditional probability of winning by switching is

P(CK2|H3,X1) = P(H3|CK2,X1)* P (CK2 X1)/ P(H3 X1)

= P(H3|CK2,X1)/[ P(H3|CK1,X1) + P(H3|CK2,X1) + P(H3|CK3,X1)]

= 1/ (1/2 + 1 + 0 )

= 2/3

Related Questions

Navigate

• Browse (All)
• Browse 2
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.