N2007 P2 Q10

A player throws three darts at a target. The probability that he is successful in hitting the target with his first throw is \(\frac18\). For each of his second and third throws, the probability of success is

• twice the probability of success on the preceding throw if that throw was successful,
• the same as the probability of success on the preceding throw if that throw was unsuccessful.

Construct a probability tree showing this information.[3]

Find

1. the probability that all three throws are successful,[2]
2. the probability that at least two throws are successful,[2]
3. the probability that the third throw is successful given that exactly two of the three throws are successful.[4]