Yahtzee Thoughts
I played a couple games of Yahtzee with family members during a recent vacation, and realized that many (perhaps most) people play the game without realizing the correct odds on certain rolls.
For example, one of the objects in the game is to roll the long straight--either 2 to 6 or 1 to 5. Suppose that you roll four of the required numbers on your first throw, say 1,2,3,4. You have one die left, but two rolls. What are the odds that you will roll the five? Obviously on the next roll it's 1 in 6 and the same applies for the final roll, so your odds are 2 in 6 or 1 in 3, right?
Nope. Sometimes it helps to look at the odds against something happening. In this case, the odds that you don't roll the 5 are 5/6 on each roll. If we multiply those two together, we can see that the odds that you won't roll the 5 are 25/36. Which means that the odds that you will roll the five are only 11/36, or slightly less than 1/3.
This is obviously counter-intuitive, so let's list all the possible rolls:
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
If you count them up, there are 11 out of rolls with at least a 5, although there are 12 5s overall. The key is that 5,5 roll; that second five is as useful as nipples on men.
There are similar mistakes in statistics on other possible rolls. For example, both my sister and I were exasperated at the number of times we'd roll three of a number, then fail to get the fourth (or fifth) of that number on the two succeeding chances. Again, it seems like the odds are 2/6 twice, or 4/6 (about 67%).
But in fact, they aren't that good. We've already seen that the odds of rolling against rolling any specific number with two dice are 25/36, so all we have to do to find out the odds against it with two rolls are to square that number. It comes out to 625/1296, so the odds of getting that fourth one are 671/1296, which is about 52%.
Suppose you are at the very end and you need to roll three or more of some particular number in order to get your bonus. What are the odds that you will do just that? Unlike in the above cases I am not able to calculate the odds directly, so I set up a spreadsheet where I generated five random numbers. If any of the numbers was a 6, I told the spreadsheet to leave it alone, but otherwise to reroll. Once again I checked for 6s and rerolled any that weren't. Overall I was a little surprised. The percentage of three 6s or better bounced around a bit, but generally was around 35.5%. The percentage of four 6s or better was around 10.4%. And the percentage of Yahtzees (five 6s) was only about 1.3%.
