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Difference between revisions of "2020 CIME II Problems/Problem 6"

(Created page with "==Problem== An infinite number of buckets, labeled <math>1</math>, <math>2</math>, <math>3</math>, <math>\ldots</math>, lie in a line. A red ball, a green ball, and a blue bal...")
 
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==Problem==
 
==Problem==
 
An infinite number of buckets, labeled <math>1</math>, <math>2</math>, <math>3</math>, <math>\ldots</math>, lie in a line. A red ball, a green ball, and a blue ball are each tossed into a bucket, such that for each ball, the probability the ball lands in bucket <math>k</math> is <math>2^{-k}</math>. Given that all three balls land in the same bucket <math>B</math> and that <math>B</math> is even, then the expected value of <math>B</math> can be expressed as <math>\tfrac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 
An infinite number of buckets, labeled <math>1</math>, <math>2</math>, <math>3</math>, <math>\ldots</math>, lie in a line. A red ball, a green ball, and a blue ball are each tossed into a bucket, such that for each ball, the probability the ball lands in bucket <math>k</math> is <math>2^{-k}</math>. Given that all three balls land in the same bucket <math>B</math> and that <math>B</math> is even, then the expected value of <math>B</math> can be expressed as <math>\tfrac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
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==Solution==
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The probability that all three balls land in box <math>2n</math> is <math>\frac{1}{64^n}</math>. This means that the probability that the three balls land in the same even box is <math>\frac{1}{64} + \frac{1}{64^2} + \ldots = \frac{1}{63}</math>. This means that the probability that all three balls land in box <math>2n</math> \emph{given that they land in the same even box}

Revision as of 14:34, 6 February 2023

Problem

An infinite number of buckets, labeled $1$, $2$, $3$, $\ldots$, lie in a line. A red ball, a green ball, and a blue ball are each tossed into a bucket, such that for each ball, the probability the ball lands in bucket $k$ is $2^{-k}$. Given that all three balls land in the same bucket $B$ and that $B$ is even, then the expected value of $B$ can be expressed as $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

The probability that all three balls land in box $2n$ is $\frac{1}{64^n}$. This means that the probability that the three balls land in the same even box is $\frac{1}{64} + \frac{1}{64^2} + \ldots = \frac{1}{63}$. This means that the probability that all three balls land in box $2n$ \emph{given that they land in the same even box}

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