Consider rolling a single, fair, six-sided die twice.  What is the probability (as a decimal) of rolling an even number on the first roll and an odd number on the second roll?

Solution:  Since the die is fair, there are six possible outcomes to a single roll,  all of which are equally likely.  The sample space for a single roll is thus: {1, 2, 3, 4, 5, 6}.  In this sample space, there are three odd numbers and three even numbers.  Therefore, on a single roll, we have:

P(odd) = 3/6 = 1/2 = 0.5

and 

P(even) = 3/6 = 1/2 = 0.5

So, the probability of rolling an even number on an single roll is equal to 0.5, which makes sense because half of the sides show an even number, and half show an odd number.  The same can be said for rolling an odd number on a single roll, for the same reason.

Consecutive rolls of a single die are independent events, because the outcome of one roll cannot influence the outcome of another roll.  This allows us to compute the probability of two consecutive rolls by simply multiplying the probabilities of each roll.  Therefore, the probability of rolling an even number followed by an odd number is then just:

P(even) x P(odd) = 0.5 x 0.5 = 0.25

and so, the overall probability of rolling an even number followed by an odd number is 0.25.  





