Question: Al, Bob and Clifford go out for coffee. They play the following game to determine which one of the three pays for all three coffees.

Round 1: First Al, then Bob, and finally Clifford flip a fair coin. If there is an “odd man” i.e. odd man with a tail while the other two show a head, or odd man with a head while the other two show tail, the odd man pays the check

Round 2: If all three show a head or if all three show a tail, repeat the sequence as in round 1.

Additional rounds are played until there is an odd man. Let X be the number of rounds until the odd man is determined. What kind of special random variable is X?

Find the follow probabilities

(a) \(\Pr \left( X=1 \right)\)

(b) \(\Pr \left( X<4 \right)\)

(c) \(\Pr \left( X\ge 8 \right)\)