MathDB

6

Part of 2023 LMT Fall

Problems(2)

2023 Fall LMT Team #6 standard 6 sided die

Source:

2/4/2024
Jeff rolls a standard 66 sided die repeatedly until he rolls either all of the prime numbers possible at least once, or all the of even numbers possible at least once. Find the probability that his last roll is a 22.
combinatorics
2023 Fall Speed p6

Source:

12/23/2023
Blue rolls a fair nn-sided die that has sides its numbered with the integers from 11 to nn, and then he flips a coin. Blue knows that the coin is weighted to land heads either 13\dfrac{1}{3} or 23\dfrac{2}{3} of the time. Given that the probability of both rolling a 77 and flipping heads is 115\dfrac{1}{15}, find nn.
Proposed by Jacob Xu
Solution. 10\boxed{10} The chance of getting any given number is 1n\dfrac{1}{n} , so the probability of getting 77 and heads is either 1n⋅13=13n\dfrac{1}{n} \cdot \dfrac{1}{3}=\dfrac{1}{3n} or 1n⋅23=23n\dfrac{1}{n} \cdot \dfrac{2}{3}=\dfrac{2}{3n}. We get that either n=5n = 5 or n=10n = 10, but since rolling a 77 is possible, only n=10n = \boxed{10} is a solution.
2023FAlLspeedCombo