Two “loaded” dice each have the property that a or a is three times as likely to appear as a , or on each roll. What is the probability that a will be the sum when the two dice are rolled?

Source: NCTM Mathematics Teacher, February 2006

**SOLUTION**

If the dice are not loaded, each number is equally likely to come up. The probability of rolling each number on a single die is . Note that .

What if the dice are loaded in favor of and ?

If the probability of rolling a , or equals , the probability of rolling a or equals .

The probability of rolling a , or on a single die equals and the probability of rolling a or on a single die equals .

The sum of occurs when two dice are rolled as follows

The probability of a sum of

**Answer**: