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Discard Dice (Posted on 2012-07-27) Difficulty: 3 of 5

In a dice game, 2 players each simultaneously cast 4 fair 6-sided dice, to form a 'hand'.

Each player then:-
(i) discards the highest-scoring die from his hand, and
(ii) scores the value of the next-highest (or highest-equal) die in his hand.

If my opponent has cast {6,4,4,3}, discarding the 6 to score 4, what is the probability that my score exceeds his?

See The Solution Submitted by broll    
Rating: 3.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution (spoiler) | Comment 3 of 4 |
(a) The probability that a single die exceeds 4 is 1/3.
     The probability that a single die does not exceed 4 is 2/3.

(b) The probability that none of 4 dice exceed 4 = (2/3)^4 = 16/81

(c) The probability that exactly one die exceeds 4 = 4*(1/3)*(2/3)^3 = 32/81

As (b) and (c) are mutually exclusive, and as these are the only way to lose or tie, the second player wins with probability 
   1 - 16/81 - 32/81 = 33/81 = 11/27

This result agrees with the previous two answers

Edited on February 18, 2017, 10:34 pm
  Posted by Steve Herman on 2012-07-27 19:36:40

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