# More Probability

We played Pig again.

- In turn, each player is handed four dice. He can choose to roll 1, 2, 3 or all 4 dice. He can roll as many times as he likes.
- Each time he rolls, he computes the sum of all the dice he rolled. That is his score.
- If he rolls one “1” at any point, his turn ends, and he loses all the points he’s scored in that round.
- If he rolls two “1”s — his turn ends, and he loses all the points he’s scored so far in the whole game.
- The first player to reach 100 points wins.

- with 1 die there are 6 (6 to the 1st power) possibilities
- with 2 dice there are 6×6 (6 to the 2nd power) possibilities or 36
- with 3 dice there are 6x6x6 (6 to the 3rd power) possibilities or 216

This time I asked the kids this question: “What are the odds of getting a “1” if you roll one die?” (That’s easy: 1:6) “What are the odds of rolling at least one “1” if you roll 2 dice?” Again, I showed them how to construct a tree diagram to show that there are 36 possible outcomes when you roll 2 dice. The odds of rolling at least one “1” are 11:36. So when you choose two dice rather than one — in hopes of getting more points — you almost double your odds of getting a “1,” which would cause you to lose your points for that round. Then I asked “What are the odds of rolling at least one “1” when you roll 3 dice, and tried to guide them through using the tree diagram to discover that there are 216 possible outcomes with 3 dice.

This was too difficult for both kids. I tried a different approach, using grids instead of tree diagrams to illustrate the possible outcomes. I’ve never seen it done this way, but I thought it would visually spotlight the multiplication & exponents involved.

with one die: 6 possibilities (1 chance out of 6 of getting a “1”):

1 | 2 |

3 | 4 |

5 | 6 |

with two dice: 36 possibilities (11 chances out of 36 of getting at least one “1”):

1 & 1 | 1 & 2 | 1 & 3 | 1 & 4 | 1 & 5 | 1 & 6 |

2 & 1 | 2 & 2 | 2 & 3 | 2 & 4 | 2 & 5 | 2 & 6 |

3 & 1 | 3 & 2 | 3 & 3 | 3 & 4 | 3 & 5 | 3 & 6 |

4 & 1 | 4 & 2 | 4 & 3 | 4 & 4 | 4 & 5 | 4 & 6 |

5 & 1 | 5 & 2 | 5 & 3 | 5 & 4 | 5 & 5 | 5 & 6 |

6 & 1 | 6 & 2 | 6 & 3 | 6 & 4 | 6 & 5 | 6 & 6 |

James and I looked at several patterns on the last chart, such as the odds of rolling doubles.

The kids “got” these tables, but they didn’t see the pattern or understand how this information could be used to work out the odds of rolling at least one “1” when three dice are rolled. I didn’t want to explain it; I want them to discover it for themselves. So I set it aside; we’ll come back to this at some point in the future.