We were in the Jolly Quizmaster the other night and saw three people playing a game of cards that we had not seen before. So we watched and worked out how the game worked.
The three players were using just ten cards, the Ace (counting one) to Ten of Hearts. Each player received three cards and the tenth and final card was placed in the middle of the table, face down.
Each player then put £1 into the kitty and card in the middle of the table was turned face up. Then they each totalled up their three cards plus the middle card, and whoever had the largest total won the game, and the £3. If two or three players had the same total they shared the money.
We could see the cards held by one of the players, but not the other two. On one hand we could see he held 9, 6 and 2, and one of our colleagues offered to buy the hand for 50p.
Should the player sell the hand or not?
Let us consider the seven possible values for the hidden card in the middle:
If it is a ten then the total is 27, and the other two players hold 1, 3, 4, 5, 7 and 8 between them. Of the ten permutations the player will win six, lose three, with one draw. So 6.5 out of 10.
Continuing:
If it is an eight then it is 3.5 out of 10,
If it is a seven then it is 3 out of 10,
If it is a five ten then it is 0.5 out of 10,
and it is 0 out of 10 for the others.
In total there are 13.5 wins out of 70 possible results and £3 * 13.5 / 70 is nearly 58p.
So the player should not sell.