Difference between revisions of "2023 SSMO Speed Round Problems/Problem 7"
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==Problem== | ==Problem== | ||
− | At FenZhu High School, <math>7</math>th graders have a 60 | + | At FenZhu High School, <math>7</math>th graders have a 60% of chance of having a dog and <math>8</math>th graders have a 40% chance of having a dog. Suppose there is a classroom of <math>30</math> <math>7</math>th grader and <math>10</math> <math>8</math>th graders. If exactly one person owns a dog, then the probability that a <math>7</math>th grader owns the dog is <math>\frac{m}{n},</math> for relatively prime positive integers <math>m</math> and <math>n.</math> Find <math>m+n.</math> |
==Solution== | ==Solution== |
Latest revision as of 14:21, 3 July 2023
Problem
At FenZhu High School, th graders have a 60% of chance of having a dog and
th graders have a 40% chance of having a dog. Suppose there is a classroom of
th grader and
th graders. If exactly one person owns a dog, then the probability that a
th grader owns the dog is
for relatively prime positive integers
and
Find
Solution
The probability that a th grader has the only dog is
and the probability for an
th grader is
Then,
The probability is thus so the
sum is