The dice game of the fair The following dice game is very popular at fairs but it is rare for two people to agree on the chances of winning the player has, so I present it as an elementary problem of probability theory.

On the board we have six boxes marked with the numbers 1, 2, 3, 4, 5 and 6. Players are invited to place as much money as they wish in any of these boxes. Three dice are then thrown and if the number that has been chosen appears on a single die, double the money bet is recovered. If the number appears on two dice, the player recovers triple the money wagered. If the number appears on three dice, the bet money is recovered four times. Obviously if the number does not appear on any of the dice, the trader keeps our money.

To clarify it with an example, suppose you bet \$ 1 on number 6. If a dice shows a 6 you get your dollar plus another dollar. If there are two dice that show 6 you get your dollar back and win two more. If all three dice show a 6 you get your dollar back and earn three more dollars.

Any player could think the following: the probability that my number appears on a dice is 1/6 but since the dice are three the odds are 3/6 that is, I will win 50% of the time therefore the game is fair .

Of course, this is what the game owner wants to be created because it is not clear that the assumption is true.

Is the game favorable to the owner or the player? How favorable is it?

Solution

Of the 216 equally possible ways in which the dice can be thrown you will win in 91 cases and lose in the other 125. So the probability of winning the same that was bet or more is 91/216 (which transformed into probability of winning the same thing that was bet is 100/216) and therefore the player's probability of losing is 125/216 = 57.87%.

If the dice always showed different numbers the game would be fair.

Suppose all the boxes were covered by a dollar bet. In each roll that showed three different numbers the merchant would win three dollars and would have to pay another three. But in the doubles the owner earns a dollar and in the triples two. In the long run for every dollar wagered by a player regardless of how he plays the money and in what amounts a loss of around 7.87 cents can be expected.

This gives the trader a profit of 7.87% on each dollar wagered.