The Two Dice Wager

A gambler presents you with an even-money wager. You will roll two dice, and if the highest number showing is one, two, three or four, then you win. If the highest number on either die is five or six, then she wins. Should you take the bet?


Two Dice

See Answer
You may be tempted to take the bet because you can calculate that your odds of winning with one die are 4 out of 6, or 2/3. What happens now that two dice are involved?

Instead of six possibilities, there are now 36. You can make a simple grid to see your odds of winning, with the six possible results for the first die forming the columns, and the six possible results for the second die forming the rows. As you can see, your odds of winning are 16 out of 36, or 4/9, so you should not take the bet.

Two Dice Probabilities

Remember, if a five or a six shows on either die, the gambler wins. So with two dice, you have to win on both at once, which is harder. The odds of winning with one die are 2/3, so another way to figure the odds of winning on both dice at once is to multiply the probabilities: 2/3 times 2/3 equals 4/9.


5 thoughts on “The Two Dice Wager”

    1. This puzzle is certainly trickier than it seems. The way the wager is phrased, if there is a 5 or a 6 on either die, the gambler wins. With two dice, the gambler will win about 5 out of 9 rolls. If you don’t believe it, you can test it with real dice.

    2. You’re not too smart, are ya? 1 dice, your odds are 2/3. 2 dice, your odds are 4/9. 3 dice, your odds are, 8/27. Aka your statement is dead wrong and actually backwards. This is because to win the bet, you need to win all the dice involved.

    1. Nope. If you don’t believe it, try it with real dice! Remember, with two dice you have to win on both of them, which lowers the probability. Think of it this way: if you flip one coin, you have a 1/2 chance of getting heads. If you flip two coins, you only have a 1/4 chance of getting heads on both of them.

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