Joint Distribution Functions/ part5

Example. A fair coin is tossed 10 times; the random variable X is the number of heads in these 10 tosses, and Y — the number of heads in the first 3 tosses. In spite of the fact that Y emerges before X it may happen that someone knows X but not Y.

Conditional probability
Given that X = 1, the conditional probability of the event Y = 0 is P ( Y = 0 | X = 1 ) = P ( Y = 0, X = 1 ) / P ( X = 1 ) = 0.7. More generally,

for x = 0, 1, 2, 3, 4, 5, 6, 7; otherwise (for x = 8, 9, 10), P ( Y = 0 | X = x ) = 0. One may also treat the conditional probability as a random variable, — a function of the random variable X,


The expectation of this random variable is equal to the (unconditional) probability,


which is an instance of the law of total probability E ( P ( A | X ) ) = P ( A ).

Thus, P ( Y = 0 | X = 1 ) may be treated as the value of the random variable P ( Y = 0 | X ) corresponding to X = 1. On the other hand, P ( Y = 0 | X = 1 ) is well-defined irrespective of other possible values of X.