Having been inspired by the recent World Cup, A approaches B with a proposition:

**A:** Hey B, let’s play a game of sudden-death penalties. If I score and you miss, you pay me $10. If you score and I miss, I’ll pay you $10. If we both score or both miss, we play another round. How about it?

**B:** That’s not a fair game! Your scoring rate against me is twice my scoring rate against yours! How about this: If you score and I miss, I’ll pay you $10, but if I score and you miss, you have to pay me double-$20. That sounds fairer!

**Should A accept B’s counter proposal?**

**The first thing we should work out is the probability of A winning the game.** If we let be the probability that A wins the game, then the probability that B wins the game is . A’s expected winnings from the game, which we will denote with , would then be determined as below:

**A should accept the counter offer if his expected winnings is positive, and reject it if his expected winnings is negative.**

As with this AMC problem, we could **play one round of the game and examine the possible states we find ourselves in**. Let the probabilities of A scoring and B scoring be and respectively. After one round, we find ourselves in one of 3 possible states:

**Case 1:** A scores and B misses.

In this case, A wins. The probability of this case happening is .

**Case 2:** A misses and B scores.

In this case, A loses. The probability of this case happening is .

**Case 3:** A & B both score or both miss.

In this case, **we find ourselves back at square one**. Hence, A has probability of winning the game. The probability of this case happening is .

Summarising the 3 cases, we have an equation for :

In the problem we are not told what the values of and are, just that . Substituting this into the expression for above:

Putting it all together in our initial expression for A’s expected winnings, we get

which is definitely non-negative since the denominator is always positive (*why?*). Hence, under the counter proposal A’s expected winnings are non-negative, so **he should accept the counter proposal**.

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