12. Let’s make a deal. In the game Let’s Make a Deal, the host asks a participant to choose one of three doors. Behind one of the doors is a big prize (e.g., a car), while behind the other two doors are minor prizes (e.g., a blender). After the participant selects a door, the host opens one of the other two doors (knowing it is not the one having the big prize). The host does not show the participant what is behind the door the participant chose. The host asks the participant to either

(a) stick with his/her original choice, or

(b) select the other of the remaining two closed doors.

Find the probability that the participant will win the big prize for each of the strategies

(a) and (b).

13. Using only the three axioms of probability, prove parts (1) and (2) of Proposition 2.4-1. (Hint. For part (1)

apply Axiom 2.4.3 to the sequence of events E1 = S and Ei = ∅ for i = 2, 3, . . ., the union of which is S. This results in the equation P(S) =

1∞i =1 P(Ei) = P(S) +

1∞i =2 P(∅).

Now complete the argument. For part (2) apply Axiom 2.4.3 to the sequence of events E1, . . . ,En, and Ei = ∅ for i = n + 1, n + 2, . . ., the union of which is ∪ni

=1Ei.

This results in the equation P(∪ni

=1Ei) =

1∞i =1 P(Ei) = P(∪ni

=1Ei) +

1∞i =n+1 P(∅). Now complete the argument.)