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Mathematics-Online problems:

# Interactive Problem 1356: Yahtzee

 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Problem Interactive Problem

The object of the game Yahtzee'' is to make as advantageous scoring combinations as possible by rolling five dice. There are, among others,
• Three-of-a-kind (three dice showing the same face and two showing different ones), e.g.
• a Small Straight (exactly four sequential dice), e.g.
.

What is the probability of a Three-of-a-kind and a Small Straight, respectively, in the very first roll?

Let us assume that you obtained the combination

,
which you would like to improve by another roll.

Determine the probability that you achieve
• altogether at least three identical faces by re-rolling three dice or
• at least four sequential faces by re-rolling two dice.
Assume that in each case of re-rolling the best possible choice is made among the dice.

Probability of
 a Three-of-a-kind: a Small Straight:
in one roll

Probability of
 at least three identical faces: at least four sequential faces:
when re-rolling

(The results should be correct to four decimal places.)