Jeremy and Leila play darts. Jeremy is bad at it ,his throws land uniformly at random on the circular dart board.
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Jeremy and Leila play darts. Jeremy is bad at it ,his throws land uniformly at random on the circular dart board.

Leila is better, hers land uniformly at random in the half-radius disc centered on the bulls-eye. The one who plays throws the dart and scores a point whenever it lands in the central quarter-radius disc.

(a) Every round, they both play. But they throw a fair coin to decide who goes first. You turn your back for a moment, and turn around to see that the first dart has scored. What's the probability that Jeremy threw that dart?

(b) Let X be Jeremy's score after 16 rounds and let Y be Leila's. Find E [X] and E [Y].

(c) You could gamble. Whichever player you bet on, after 16 rounds, you get that player's score times $10, minus a $20 gambling fee. If you bet on Jeremy once for every two times you bet on Leila, what is your expected net payout?

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