# Two Dice, Same Score

You roll a pair of six-sided dice and sum them to get a total.

Your good friend does the same. What are the chances that you both get the same total?

What’s your gut feeling of the answer?

## Analysis

Let’s first look at one die. Whatever you roll, there’s a 1/6 chance your friend rolls the same number. Easy.

For two dice there’s a distribution of possible sums from 2–12. Here’s a graph. There are 36 possible combinations the two dice.

TOTAL | Probability | |
---|---|---|

2 | 1/36 | |

3 | 2/36 | |

4 | 3/36 | |

5 | 4/36 | |

6 | 5/36 | |

7 | 6/36 | |

8 | 5/36 | |

9 | 4/36 | |

10 | 3/36 | |

11 | 2/36 | |

12 | 1/36 |

The probability of getting the same total is the probability of you both rolling a count of two, or the probability of you both rolling three, or … both rolling twelve.

Pr = Pr(both rolled 2) + Pr(both rolled 3) + Pr(both rolled 4) + … + Pr(both rolled 12)

*The percentage chance that the total of two pairs of dice will sum to the same value is just over 11%*

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## More Dice

We can expand this to more dice and look at the binomial coefficients to calculate exact answers but this afternoon I was feeling lazy, so just wrote a few lines of code to brute force the results using a Monte-Carlo simulation. Here are the results for up to 32 dice.

Again, before I did this, I had some simple gut feeling about the results, and the central limit theorem tends to smooth out the summation distribution as the number of dice increases, but the percentage chance of a match score falls off a lot slower than I would have predicted.