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In a game of Yathzee, five balanced dice are rolled simultaneously. Find the probabilities of getting:

a) two pairs

b) three of a kind

C) a full house (three of a kind and a pair)

D) four of a kind

Please help me with this question. I don't understand this game.

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  • $\begingroup$ Is the question about probability and combinatorics, or rather about the rules of yathzee? $\endgroup$
    – ajotatxe
    May 16, 2015 at 9:39
  • $\begingroup$ Hint: Two pair mean you get a roll like $(2,2,3,3,6).$ Full House means something like $(3,4,4,3,4).$ Four of kind is something like $(2,5,2,2,2).$ The odds of three of kind can be computed as follows: $\frac{6 \cdot C(5,3) \cdot 5 \cdot 4}{ 6^5}$ $\endgroup$ May 16, 2015 at 9:39
  • $\begingroup$ This provides a good explanation bgsu.instructure.com/courses/901773/pages/… $\endgroup$ Jul 15, 2020 at 20:51

2 Answers 2

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I will do A for you, and then I hope you can figure the rest yourself. We first need to choose the number of dots for the pairs. This can be done in ${6 \choose 2} =15$ ways. Then we want to choose a number of dots for the single, this can be done in 4 ways. The number of reorderings of these is $\frac{5!}{2!^2}=30$. This gives in total $15\cdot4\cdot30=1800$ ways to get two pairs. The total number of dice throws is $6^5=7776$. So the odds of getting two pairs is $\frac{1800}{7776}=0,231$.

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The dice are normal six-sided dice. The rest seems rather self-explanatory; for detailed rules of the game see its Wikipedia page.

But for some detail:

  • two pairs would mean you roll, say, $2,2,5,5,6$ so a pair of $2$s and a pair of $5$s.

  • three of a kind means you roll the same number three times so $4,4,4,5,6$ for instance.

  • full house means a pair of one number and three of a kind of another number, so $(1,1,1, 5, 5)$ for example.

  • four of a kind means you roll the same number four times so $4,4,4,4,6$ for instance.

Note that there is no requirement the numbers appear in this order in the rolls I just grouped them for easy reading.

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