Probability - Probabilities of Successive Events

Ava and Ben take turns throwing a dart at a target. The probability that Ava hits the target on any throw is \(\dfrac{2}{5}\), while the probability that Ben hits it on any throw is \(\dfrac{1}{3}\). All throws are independent. A player wins as soon as that player hits the target. Ava throws first.

1. Show that the probability that both players miss in one pair of consecutive throws, one by Ava followed by one by Ben, is \(\dfrac{2}{5}\).[1]
2. Find the probability that Ava wins the game.[3]

In a second game, a biased coin is used to choose who throws first. Ben is twice as likely as Ava to throw first. The same hit probabilities and rules apply.

3. Find the probability that Ava wins the second game.[3]