Max Value

You roll three fair six-sided dice. What is the probability that the highest value rolled is exactly four?

Answer

To find the probability that the highest value rolled is exactly four, we calculate the probability that all dice show a value less than or equal to four and subtract the probability that all dice show a value less than or equal to three. This can be represented as:

P(\text{max} = 4) = P(\text{max} \leq 4) - P(\text{max} \leq 3)

Calculating these probabilities gives:

P(\text{max} \leq 4) = \left(\frac{4}{6}\right)^3, \quad P(\text{max} \leq 3) = \left(\frac{3}{6}\right)^3

Thus:

P(\text{max} = 4) = \left(\frac{4}{6}\right)^3 - \left(\frac{3}{6}\right)^3

Simplifying:

P(\text{max} = 4) = \frac{64}{216} - \frac{27}{216} = \frac{37}{216}

Notes and comments

Comment 1: We subtract $P(\text{max} \leq 3)$ from $P(\text{max} \leq 4)$ because $P(\text{max} \leq 4)$ includes all sequences where the highest value is 4 or less, including those where the highest value is 3 or less (sequences such as 1, 1, 1 or 2, 1, 1 or 3, 2, 3). By subtracting $P(\text{max} \leq 3)$ , we isolate the probability that the highest value is exactly 4.

Comment 2: The general formula for the probability that the highest value rolled is exactly $$ k $$ out of $$ n $$ throws is given by:

P(\text{max} = k) = \frac{k^n - (k-1)^n}{6^n}.