Probability in Yahtzee

Date: 03/20/2002 at 20:50:09
From: Julie
Subject: Probability in Yahtzee

1. What is the probability of rolling two of a kind three times? 
(There are five dice and you can start over every time you get your 
two of a kind.)

2. What is the probability of rolling four of a kind?

3. What is the probability of rolling a yahtzee (all five the same) 
on the first roll? And again on the second?

I don't understand probability this complex. 
Thank you for the help.

Date: 03/20/2002 at 22:46:39
From: Doctor Twe
Subject: Re: Probability in Yahtzee

Hi Julie - thanks for writing to Dr. Math.

Probabilities in Yahtzee are not straightforward because the game 
involves decision making on the players' part. If, for example, we 
roll two 2's, two 3's, and a 5; do we keep both pair? Do we re-roll 
one pair (along with the 5) hoping to score a 3-of-a-kind, 4-of-a-kind 
or yahtzee? 

Our optimal strategy will also depend on which combinations I have 
filled in already. If, for example, I get the above roll but I've 
already filled in my 2's and my full house, I'd be more likely to 
re-roll them hoping for more 3's. If, on the other hand, my 2's are 
still available and so is the full house, I'd be better off only 
re-rolling the 5, hoping for either a 2 or 3. (At worst, I'd just 
score 4 points in the 2's place, only 2 points short of the "target 
value.")

>1. What is the probability of rolling two of a kind three times? 

Do you mean 3 times in a row? Or 3 times over the course of a game? 

>2. What is the probability of rolling four of a kind?

As explained above, this depends on the game situation and the 
player's strategy on re-rolling.

>3. What is the probability of rollong a yahtzee (all five the same) 
>on the first roll? And again on the second?

The probability of rolling a yahtzee on any particular (single) roll 
(no re-rolling) is:

     # ways to roll yahtzee       6       1       1
     -----------------------  =  ---  =  ---  =  ----  ~=  0.07716%
     # possible combinations     6^5     6^4     1296

The probability of rolling two consecutive yahtzees is:

     P(2_yahtzees) = P(yahtzee) * P(yahtzee)
                   = P(yahtzee)^2
                   = (1/1296)^2
                   = 1/1,679,616
                  ~= 0.00005954%

For more on Yahtzee probability, see:

   Yahtzee Probabilities and a State Matrix
   http://mathforum.org/dr.math/problems/ryan12.30.98.html   

I found that entry by typing "yahtzee" (without the quotation marks) 
into our Ask Dr. Math archive searcher at:

   http://mathforum.org/mathgrepform.html   

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/