Editing Conditional expectation (section)

=== Example 1: Dice rolling ===
Consider the roll of a fair die and let ''A'' = 1 if the number is even (i.e., 2, 4, or 6) and ''A'' = 0 otherwise. Furthermore, let ''B'' = 1 if the number is prime (i.e., 2, 3, or 5) and ''B'' = 0 otherwise.
{| class="wikitable"
|-
!  !! 1 !! 2 !! 3 !! 4 !! 5 !! 6
|-
| A || 0 || 1 || 0 || 1 || 0 || 1
|-
| B || 0 || 1 || 1 || 0 || 1 || 0
|}

The unconditional expectation of A is <math>E[A] = (0+1+0+1+0+1)/6 = 1/2</math>, but the expectation of A ''conditional'' on ''B'' = 1 (i.e., conditional on the die roll being 2, 3, or 5) is <math>E[A\mid B=1]=(1+0+0)/3=1/3</math>, and the expectation of A conditional on ''B'' = 0 (i.e., conditional on the die roll being 1, 4, or 6) is <math>E[A\mid B=0]=(0+1+1)/3=2/3</math>. Likewise, the expectation of B conditional on A = 1 is <math>E[B\mid A=1]= (1+0+0)/3=1/3</math>, and the expectation of ''B'' conditional on ''A'' = 0 is <math>E[B\mid A=0]=(0+1+1)/3=2/3</math>.