Question

Reasoning about Events
Alex says that if he rolls an odd number on the first number cube, it is more likely that he will roll an even number on the second number cube. Is he correct? Explain.

Answer 1

No — Alex is incorrect. Rolls of a fair number cube are independent, so the first roll doesn't change the chance for the second.

Calculation: P(second is even) = 3/6 = 1/2.
P(second is even | first is odd) = P(first odd and second even) / P(first odd) = (1/2·1/2) / (1/2) = 1/2.

Or count outcomes: there are 36 ordered pairs. If the first is odd (3 choices) and the second is even (3 choices) there are 3·3 = 9 favorable pairs out of 18 pairs with first odd, giving 9/18 = 1/2. Thus the probability is the same either way.