Six, Five, Four: A Probability Game

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Analyze this simple betting game with your fourth grade student. Discuss probability as you work to figure out whether the odds are in the dealer's or the player's favor. All you need is a set of three dice, and you can start playing! Play several times and look for patterns in the game. Who is most likely to win? Who is least likely to win? Take a close-up look at each round of the game. If you are interested in going a step further and looking into the exact calculations, read the "Probability Discussion" posted at the bottom of the page.

Term to Know:

probability: a measure of how likely it is an event will occur

What You Need:

Three ordinary dice (Make your own.)

What You Do:

Select a dealer and a player. The object of the game is for the player to get 6, 5, and 4 on three rolls of the dice.
The player plays three rounds, rolling the dice in each round. The goal is to get one 4, one 5, and one 6 by the end of the third round. If one of these numbers comes up in the first or second round, they should hold that dice and not roll it again.
If the player succeeds in rolling a 4, 5, and 6, they earn a point. If not, the dealer earns a point.
Switch roles and play again.
After playing several times, discuss what you think the probability, or odds, of the game are. Is the game fair? Who has the better odds, the dealer or the player?

Probability Discussion:

If you are interested in the logic behind the odds, or probability, of winning, read on.

Start by figuring out the odds of rolling just a 6. Is it likely, unlikely, probable...? Well, you have at least 5 chances of rolling a 6. Why 5? Because you may roll a 4 and a 5 in the first round, limiting your dies for the subsequent rounds to 1. So, you could roll 3 dice in the first round, 1 die in the second round, and 1 die in the third round. With at least 5 chances to roll a 6, the odds look pretty good. Now, let's look at all three numbers...
In the first round, it's very likely you'll roll a 4, 5, or 6. Why? Because these are exactly 1/2 (or 3/6) of the numbers on a dice. Multiply that by 3 (for 3 dice), and chances are you will roll 1 1/2 of those numbers. (Now you're thinking, wait! You can't roll 1/2 of a number! Well, you're right. You can't. What these odds mean is you are likely to roll at least 1 of your numbers, and half the time you should roll 2 of them. This averages out to 1 1/2.)
In the second round, the odds get a little bit tricky. Let's say you rolled just one of your numbers in the first round. This means, you have 2 dice to roll and 2 numbers to find. The odds of finding one of those remaining numbers is 1/3 (or 2/6). It is unlikely you will find either of your numbers in this round, but there is a chance. If you do find one of your numbers, you've just lowered your chances of finding the last number in the last round, with one dice, to 1/6. That is very unlikely.
But - what if you rolled two of your numbers in the first round? (After all, that will probably happen 1/2 the time.) You would have 1 dice to roll and one number to find. Your odds are pretty low, as you now have a 1/6 chance in finding that number. You do get another roll in the third round, but you'll still only have a 2/6 (or 1/3) chance of finding it total.
The odds do look pretty good at first, don't they? However, as you progress through the rounds, you'll realize the odds are in the dealer's favor. It's probably not a good idea to bet money on this game!