SHREWD PRISONER'S DILEMMA

Designed by Engin A. Sungur

QUESTION

Because of a prisoner's constant supplication, the king grants him this favor. He is given 2N balls, which differ from each other only in that half of them are green and half are red. The king instructs the prisoner to divide up the balls between two identical urns. One of the urns will then be selected at random and the prisoner will be asked to choose a ball at random from the urn chosen. If the ball turns out to be green, the prisoner will be freed.

How should he distribute the balls in the urn to maximize his chances of freedom?

LEARNING OBJECTIVES

To learn how to formulate a problem probabilistically. After completing this activity you should be able define compound events of interest, understand the meaning of conditional probability and law of total probability.

ACTIVITY

WARMUP

Suppose that the prisoner is given 12 balls; 6 green and 6 red.

Let g be the number of green balls and r be the number of red balls in the first urn.

What will be the number of green and red balls in the second urn?

1. What do you think that the best strategy should be? That is how would you distribute the balls into these urns?

URN 1
URN 2
Number of Green
Number of red
Number of green
Number of red
Total Number of Balls in Urn 1= Total Number of Balls in Urn 2=

2. Let

F="selecting Urn 1"

E="selecting a green ball".

Define the following events:

Fc=
Ec=
E|F=
E| Fc=
F|E=
F|Ec=

3. Determine the following probabilities for the following two cases

g=3 and r=3

P(F)=
P(Fc)=
P(E|F)=
P(E| Fc)=
P(Ec|F)=
P(Ec|Fc)=

g=1 and r=0

P(F)=
P(Fc)=
P(E|F)=
P(E| Fc)=
P(Ec|F)=
P(Ec|Fc)=


4. Complete the following probability trees. (To see the answer move the cursor on the figure and do not click the mouse.)

For g=3 and r=3

P(E)=?=

For g=1 and r=0

P(E)=?=

5. Which one of the above two cases would you choose?

Would you change your answer to the question 1?

MAIN ACTIVITY

1. Given that there are g green and r red balls in the first urn, complete the following table:


URN 1
URN 2
Number of Green
Number of red
Number of green
Number of red

Total Number of Balls in Urn 1=
Total Number of Balls in Urn 2=

2. Complete the following probability tree. (To see the answer move the cursor on the figure and do not click the mouse.)

P(E)=?=

3. The formula for P(E) contains g, r, and N. From prisoner point of view which one(s) of these is(are) fixed and which one(s) is(are) variable?

4. Let P(E)=f(g,r). How would you find the maximum of this function of two variables. (Note that the maximum need not occur at an interior point of the domain.)

EXTENSIONS

Suppose that the prisoner is given 6N (in general 2kN) balls half of them green half of them red. How should he distribute them into 3 urns (in general k urns) to maximize his chances of freedom?