Similar to the definition of the probability mass function, we can define the conditional probability mass function. The conditional probability mass function of random variable X conditioned on an event A is defined as

_pX|A(a | A) = P(X = a|A).

Similarly, we can define the conditional expectation of X in the following way:

E(X|A) = a _apX|A(a | A)

(a) (6 marks) For an event A, prove that E(X) = E(X|A)P(X A) + E(X|A^c )P(A^c ).

Now, consider the following game two players are playing. Each player has an unbiased die (numbers 1 to 6 on the sides of the die and the chance of observing each is 1/6). They roll their dice . The person who has a larger number wins the game. If they see the same numbers, the game will be a tie.

(b) (10 Marks) Let B denote the event that the first player won the second player, and X denote the number the first player has observed. Calculate the conditional probability mass function _pX|B(a | B)

(c) (5 Marks) Calculate E(X|B).

(d) (5 marks) Can you use the idea of repeated experiments to intuitively explain how E(X|B) can be obtained by averaging numbers that are obtained from repeating a random experiment (similar to what we did for explaining the importance of the mean). You can use the game that you worked on earlier in this question