Difference between revisions of "2021 Fall AMC 10A Problems/Problem 10"
MRENTHUSIASM (talk | contribs) (Created page with "==Problem== A school has <math>100</math> students and <math>5</math> teachers. In the first period, each student is taking one class, and each teacher is teaching one class....") |
MRENTHUSIASM (talk | contribs) (→Solution) |
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The formula for expected values is <cmath>\text{Expected Value}=\sum(\text{Outcome}\cdot\text{Probability}).</cmath> | The formula for expected values is <cmath>\text{Expected Value}=\sum(\text{Outcome}\cdot\text{Probability}).</cmath> | ||
We have | We have | ||
− | < | + | <cmath>\begin{align*} |
t &= \frac15\cdot50 + \frac15\cdot20 + \frac15\cdot20 + \frac15\cdot5 + \frac15\cdot5 \\ | t &= \frac15\cdot50 + \frac15\cdot20 + \frac15\cdot20 + \frac15\cdot5 + \frac15\cdot5 \\ | ||
&= \frac15\cdot(50+20+20+5+5) \\ | &= \frac15\cdot(50+20+20+5+5) \\ | ||
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&= 25 + 4 + 4 + 0.25 + 0.25 \\ | &= 25 + 4 + 4 + 0.25 + 0.25 \\ | ||
&= 33.5. | &= 33.5. | ||
− | \end{align*} | + | \end{align*}</cmath> |
Therefore, the answer is <math>t-s=\boxed{\textbf{(B)}\ {-}13.5}.</math> | Therefore, the answer is <math>t-s=\boxed{\textbf{(B)}\ {-}13.5}.</math> |
Revision as of 18:18, 22 November 2021
Problem
A school has students and teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are and . Let be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is ?
Solution
The formula for expected values is We have Therefore, the answer is