posted
That is something we asked ourselves the other night as we were having a game. It happens - you roll all 5 dice and get a yahtzee.

So, I was wondering if there are any mathematicians out there who can tell me what the odds are of doing that. Getting the same value on all 5 dice in one roll. Surely there must be some way of calculating that possibility?

I'm new to the message boards on this site, so I hope I'm in the right board.

Thanks.
Posts: 2 | From: Mackay, Australia | Registered: Dec 2005
| IP: Logged |

Let's assume you're throwing all of them sequentially, rather than at the same time. The calculations are easier to explain (though it's the identical answer - just easier to conceptualize).

First die - doesn't matter what you throw, because we're going to calculate what it would take to get the other four to match it. This way, we will calculate the odds for ANY Yatsee (any of the six possible ones), not just a single one.

Second die - has a 1/6 chance of matching the first one.

Third die - has a 1/6 chance of matching the first one, and a 1/6 * 1/6, or 1/36 chance of matching both of the first ones.

Continuing on, the chance that the fifth die will match all of the previous four dice is 1/6 * 1/6 * 1/6 * 1/6, or 1/1296.

Therefore, the odds of getting a Yatsee in a single throw is 1/1296, or about 0.08%.

Glad to be of service, DV!

-------------------- "I'm singing and deranged!" Posts: 239 | From: Georgia | Registered: Dec 2004
| IP: Logged |

posted
So the odds of rolling a yahtzee of any particular number in one roll are 1/1296 * 1/6 = 1/7776 or about .012%, correct?

To complicate the question further, what are the odds of scoring a yahtzee on one's turn (3 rolls)?

Do you sum the odds for all the particular ways that it could happen? E.g. the odds of rolling on the first throw (1/1296)+ the odds of having two match on the first throw then throwing all three matches on the second + etc.
Posts: 741 | From: Big Bend, Texas | Registered: May 2004
| IP: Logged |

quote:Originally posted by Myshkin: So the odds of rolling a yahtzee of any particular number in one roll are 1/1296 * 1/6 = 1/7776 or about .012%, correct?

To complicate the question further, what are the odds of scoring a yahtzee on one's turn (3 rolls)?

Do you sum the odds for all the particular ways that it could happen? E.g. the odds of rolling on the first throw (1/1296)+ the odds of having two match on the first throw then throwing all three matches on the second + etc.

Question 1: yep: (1/6) ^ 5

Question 2: you have to add up a whole bunch of possibilities. You *might* roll all five dice the same on your first roll: The initial subtotal is: (1/6) ^ 4

If not, then you have to start breaking down individual cases.

Case 1: you rolled four of a kind. Unfortunately, there are five different ways to do this, so the odds are 5 * (1/6) ^ 3. (Call this X4.) You might match it up with your next roll, so add X4 * 1/6 to the running subtotal. However, you might miss, and then catch it on your second try. So add X4 * 5/6 * 1/6 to the subtotal.

etc. etc.

Lotta cases.

Silas
Posts: 16801 | From: San Diego, CA | Registered: Sep 2000
| IP: Logged |

quote:Originally posted by Myshkin: To complicate the question further, what are the odds of scoring a yahtzee on one's turn (3 rolls)?

The Wizard of Odds has calculated the odds of rolling a Yahtzee in 3 rolls to be 4.6028643%.
Posts: 306 | From: Tacoma, WA | Registered: Sep 2005
| IP: Logged |