In a game of Yatzy players take turns rolling 5 dice. Various combinations earn

Question

In a game of Yatzy players take turns rolling 5 dice. Various combinations earn points. After a roll, players choose which dice to keep and which to roll again. Players may roll some or all of the dice up to a total three times 011 a turn after which a score (or a zero) is entered into a score box. The game ends when all score boxes are used. The player with the highest total score wins the game. Here we consider the simpler version in which players have only one roll of the 5 dice. What is the probability of rolling (a) Five of a kind. (b) A large straight that is 2, 3, 4, 5, 6. (c) A full house which is three of a kind and two of another (such as 5, 5, 5, 2, 2). (d) Three of a kind. (e) Two of a kind.

Explanation / Answer

There are 5 dice in one roll, possible number of outcomes are 6^5 = 7776

(a) Five of a kind is possible when all dice will show the same number, this is possible in 6C1*(1/6)^5

Hence required probability is 6*(1/6)^5 = 0.000772

(b) Probability of a large straight is (1/6)^5 = 0.000129

(c) required probability = 6C1*(1/6)^3 * 5C1*(1/6)^2 = 6*(1/6)^3 * 5*(1/6)^2 = 0.003858

(d) required probability = 6C1*(1/6)^3 * 5C1 * (1/6) * 5C1 * (1/6) = 0.01929

(e) required probability = 6C1*(1/6)^2 * 5C1*((1/6)^3) = 0.003858

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