Hidden Die

Jillian throws four 6-sided, fair dice on a table. One of the dice rolls under the table where it is hidden from her and the other three show the following result:

Before retrieving the hidden die, she says, "Since the most probable result when throwing four dice is 14 14 , I think the most probable value the hidden die will show is 1. 1. "

Is she correct?

Yes No

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2 solutions

Romain Bouchard
Jan 1, 2018

The four throws are independent so the result of the hidden die could be any value within { 1 , 2 , 3 , 4 , 5 , 6 } \{1,2,3,4,5,6\} with the same probability of 1 6 \frac{1}{6} .

Probabilities change when more information becomes available!

Once we know that the first three dice have a total of 13, the remaining (conditional) distribution is P = 1 6 P = \tfrac16 for sums of 14 through 19, and P = 0 P = 0 for any other outcomes.

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