Card Game Quiz Puzzle Answer

We were enjoying another quiet pint in the Jolly Quizmaster, reminiscing about the Name That Tune incident. On the next table two men were preparing to play a game of cards, and while one went to the bar for drinks, the other explained the game and his strategy for winning it.

The game was simplicity itself; one player made an evens bet of one pound on whether the top two cards in the pack were the same colour or different colours. The other player then turned over the top two cards, and the pound was won and lost on the basis. Play continued through the pack in this way, and then the two players swapped roles.

Our friend explained that he was going to bet on the cards being different, as once the top card had been turned there were 25 of the same colour and 26 of the other colour.

His friend returned, play started, and it being such a basic game we rather lost interest. Until we heard our friend make an exasperated remark, and we enquired as to the problem.

He explained that he had lost every hand so far. Naturally we sympathised, and then asked if he was going to continue with his strategy. He looked even more perplexed, and replied that it had reached the point where both bets had the same probability.

How much money had he lost?

Let there be r red cards left and b black cards. The probability of turning two red cards is

$$ \frac{r(r - 1)}{(r + b)(r + b -1)} $$

and of turning two black cards

$$ \frac{b(b - 1)}{(r + b)(r + b - 1)} $$

and so the probability of two cards of the same colour is the sum of these

$$ \frac{r(r - 1) + b(b - 1)}{(r + b)(r + b - 1)} $$ 

Likewise the probability of two cards of different colours is

$$ \frac{2rb}{(r + b)(r + b - 1)} $$

As the two probabilities are the same we have

$$ r^2 - r + b^2 - b = 2rb $$

$$ (r - b)^2 = r + b $$

Now r + b must be even as the cards are removed in pairs and so it must be an even square less than 52. That is, 4, 16 or 36.

If r + b is 36 then r - b is 6, and so r is 21 and b is 15.

If r + b is 16 then r - b is 4, and so r is 10 and b is 6.

If r + b is 4 then r - b is 2, and so r is 3 and b is 1.

However both r and b must be even, as either two red cards or two black cards have been removed on each hand. So the only answer is ten red cards and six black cards (or the other way round) are left in the pack.

There are 36 cards already removed from the pack, which is 18 hands. Our friend is £18 down - no wonder he's exasperated!


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