Yahtzee Question - Odds of Rolling a Yahtzee

[TomCollins]

Just want to confirm the correct answer. For those unfamiliar, you get 5 dice. You roll them. You can choose to keep each die or roll it again. Repeat one more time. If you get 5 of the same pips, you get a yahtzee.

Perfect strategy appears to be keep the one you have the most of, and roll the others. If you have 5 distinct dice, roll all again. (Don't think it matters if you roll all again or keep any one).

Also, before you do the math, give a guess on how hard it is to get a yahtzee. I'll guess its around 50-1.

[TomCollins]

My answer in white:

<font color="white"> 3.15% chance of getting a Yahtzee, or 31-1. </font>

[AaronBrown]

It's about 1 chance in 22.

100,194,336 in 2,176,782,336 if you like things exact.

It's a tedious cacluation, but not difficult. There are 6^4 = 1,296 outcomes of the first roll (you only need to roll four of the five dice, you can set the fifth to any number you want). 1 of them gives you your Yahtzee, 25 give you a 4-match, and so one. If you get a Yahtzee on any roll, you'll have a Yahtzee at the end. If you get a four match, you have one chance in 6 of converting it to a Yahtzee on the next roll. Keep working this way and you'll get it. I used Excel.

[BettyBoopAA]

That's looks about right, the wizard of odds calculates it at 4.60% (1 in 21.6)

http://www.wizardofodds.com/askthewizard/numbered/askthewizard125.html

[TomCollins]

I found my mistake. Suppose I roll 4 dice. I "improve" even if I don't match the number I held very often.