Amy, Ben, and Charlie repeatedly take turns tossing a fair die in the given order . What is the probability that Charlie will be the first one

to toss a ?

Source: NCTM Mathematics Teacher, December 2005

**SOLUTION**

The possible events that Charlie is the first one to toss a are

The probability of tossing a is and the probability of tossing anything else is .

Probability of

Probability of

Probability of

Probability of

Let be the probability that Charlie is the first one to toss a

The right hand side is the sum of the infinite geometric series with the first term and the common ratio .

**Answer**:

**Alternative solution**

Let be the probability that Charlie is the first one to toss a . Charlie can do this in two ways:

(1) both Amy and Ben fail to toss a and Charlie tosses a

probability

(2) all three players fail to toss a and Charlie is the first one to toss a from that point. The probability of Charlie winning from that point is , since whether from that point or from the beginning the situation is the same, i.e. Charlie is the first one to toss a 5.

probability

Thus,