# Probability of profit.

Consider this game:

Person A has 4 coins of 1 pound. Person B has 2 coins of 2 pounds. They are once throwing their own coin. Head-the owner keeps the coin. Tails- the owner gives this coin to the player (A/B). (But this coin doesn't take part(later) in the game). At the end of the game, they count the money. Do A and B have equal chances of getting a profit?

At first, I tried to write out all the possibilities. But this is not effective.

All we care about is the number of heads and tails each player throws.

For $$A$$ there are $$5$$ cases, as $$A$$ might throw from $$0$$ to $$4$$ tails. The probabilities are $$(.0625,.25,.375,.25,.0625)$$

For $$B$$ there are $$3$$ cases, as $$B$$ might throw from $$0$$ to $$2$$ tails. The probabilities are $$(.25,.5,.25)$$

Let's describe an outcome of the game as a pair $$(a,b)$$ where $$A$$ throws $$a$$ Tails and $$B$$ throws $$b$$ Tails. There are $$5\times 3=15$$ possible outcomes.

$$A$$ turns a profit in the scenarios $$(0,1),\,(0,2),\, (1,1),\,(1,2),\, (2,2),\,(3,2)$$

The probabilities of those cases sum to $$\boxed {.390625}$$

$$B$$ turns a profit in the scenarios $$(1,0),\,(2,0),\,(3,0),\,(4,0),\,(3,1),\,(4,1)$$ The probabilities of those cases also sum to $$\boxed {.390625}$$

Just as a consistency test, ties occur in the scenarios $$(0,0),\,(2,1),\,(4,2)$$ The probabilities of those cases sum to $$\boxed {.21875}$$ and the three total probabilities do indeed sum to $$1$$.

Person A throws 4 coins, and they has a $$50\%$$ chance of keeping each one, so they expect, on average, to end the game with $$4\cdot0.5=2$$ coins, worth 2 pounds. Similarly, Person B throws 2 coins, also with a $$50\%$$ chance of keeping each one, and so they expect to end the game with $$2\cdot0.5=1$$ coin, worth 2 pounds. So they both expect to end the game with 2 pounds worth of coins, meaning they each have equal chances to gain a profit from the game.

• It's not obvious at least not to me, why saying the game has expectation $0$ for each means that they have the same chance of getting a profit. That is not true generally (though it is in this case)...imagine a game in which I have a $\frac 34$ chance of getting $100$ from you and you have a $\frac 14$ chance of getting $300$ from me. That has expectation $0$ for both of us but I have a much greater chance of getting a profit. – lulu Jan 8 at 13:34