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A fair six-sided die is rolled 5 times. Let p be the probability that…
Question
A fair six-sided die is rolled 5 times. Let p be the probability that the maximum value obtained over the 5 rolls is exactly 5 and the minimum value obtained is exactly 2. Write p = m/n in lowest terms and find m+n.
✓ Verified answer: 1391checked by our engine — not a guess
Step-by-step solution
Use inclusion-exclusion on the event {all rolls lie in [2,5]} with corrections for the max being <5 or min being >2.
Equivalently count sequences in {2,3,4,5} that use the value 5 at least once and the value 2 at least once.
Let f(S) = number of 5-roll sequences with all entries in set S. f = |S|^5.
Let A = sequences with all rolls in {2,3,4,5} (so min>=2, max<=5): 4^5 = 1024.
We need those that also achieve max = 5 (use a 5) and min = 2 (use a 2).
By inclusion-exclusion within {2,3,4,5}:
N = (all in {2,3,4,5}) - (no 5: all in {2,3,4}) - (no 2: all in {3,4,5}) + (no 5 and no 2: all in {3,4}).
N = 4^5 - 3^5 - 3^5 + 2^5 = 1024 - 243 - 243 + 32 = 570.
Total outcomes = 6^5 = 7776.
p = 570/7776 = 95/1296.
m+n = 95 + 1296 = 1391.
Final answer1391
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