# Difference between revisions of "2020 AMC 10B Problems/Problem 18"

(→Problem) |
(→Solution) |
||

Line 7: | Line 7: | ||

==Solution== | ==Solution== | ||

− | Let <math>R</math> denote that George selects a red ball and <math>B</math> that he selects a blue one. | + | Let <math>R</math> denote that George selects a red ball and <math>B</math> that he selects a blue one. Now, in order to get <math>3</math> balls of each color, he needs <math>2</math> more of both <math>R</math> and <math>B</math>. |

+ | |||

+ | There are 6 cases: <math>RRBB, RBRB, RBBR, BBRR, BRBR, BRRB</math> (we can confirm that there are only <math>6</math> since <math>\binom{4}{2}=6</math>). However we can clump <math>RRBB + BBRR</math>, <math>RBRB + BRBR</math>, and <math>RBBR + BRRB</math> together since they are equivalent by symmetry. | ||

+ | |||

+ | |||

+ | <math>\textbf{CASE 1: }</math> <math>RRBB</math> and <math>BBRR</math> | ||

+ | |||

+ | Let's find the probability that he picks the balls in the order of <math>RRBB</math>. | ||

+ | |||

+ | The probability that the first ball he picks is red is <math>\frac{1}{2}</math>. | ||

+ | |||

+ | Now there are <math>2</math> reds and <math>1</math> blue in the urn. The probability that he picks red again is now <math>\frac{2}{3}</math>. | ||

+ | |||

+ | There are <math>3</math> reds and <math>1</math> blue now. The probability that he picks a blue is <math>\frac{1}{4}</math>. | ||

+ | |||

+ | Finally, there are <math>3</math> reds and <math>2</math> blues. The probability that he picks a blue is <math>\frac{2}{5}</math>. | ||

+ | |||

+ | So the probability that the <math>RRBB</math> case happens is <math>\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{1}{4} \cdot \frac{2}{5} = \frac{1}{15}</math>. However, since the <math>BBRR</math> case is the exact same by symmetry, case 1 has a probability of <math>\frac{1}{15} \cdot 2 = \frac{2}{15}</math> chance of happening. | ||

+ | |||

+ | <math>\textbf{CASE 2: }</math> <math>RBRB</math> and <math>BRBR</math> | ||

+ | |||

+ | Let's find the probability that he picks the balls in the order of <math>RBRB</math>. | ||

+ | |||

+ | The probability that the first ball he picks is red is <math>\frac{1}{2}</math>. | ||

+ | |||

+ | Now there are <math>2</math> reds and <math>1</math> blue in the urn. The probability that he picks blue is <math>\frac{1}{3}</math>. | ||

+ | |||

+ | There are <math>2</math> reds and <math>2</math> blues now. The probability that he picks a red is <math>\frac{1}{2}</math>. | ||

+ | |||

+ | Finally, there are <math>3</math> reds and <math>2</math> blues. The probability that he picks a blue is <math>\frac{2}{5}</math>. | ||

+ | |||

+ | So the probability that the <math>RBRB</math> case happens is <math>\frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{2} \cdot \frac{2}{5} = \frac{1}{30}</math>. However, since the <math>BRBR</math> case is the exact same by symmetry, case 2 has a probability of <math>\frac{1}{30} \cdot 2 = \frac{1}{15}</math> chance of happening. | ||

+ | |||

+ | <math>\textbf{CASE 3: }</math> <math>RBBR</math> and <math>BRRB</math> | ||

+ | |||

+ | Let's find the probability that he picks the balls in the order of <math>RBBR</math>. | ||

+ | |||

+ | The probability that the first ball he picks is red is <math>\frac{1}{2}</math>. | ||

+ | |||

+ | Now there are <math>2</math> reds and <math>1</math> blue in the urn. The probability that he picks blue is <math>\frac{1}{3}</math>. | ||

+ | |||

+ | There are <math>2</math> reds and <math>2</math> blues now. The probability that he picks a blue is <math>\frac{1}{2}</math>. | ||

+ | |||

+ | Finally, there are <math>2</math> reds and <math>3</math> blues. The probability that he picks a red is <math>\frac{2}{5}</math>. | ||

+ | |||

+ | So the probability that the <math>RBBR</math> case happens is <math>\frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{2} \cdot \frac{2}{5} = \frac{1}{30}</math>. However, since the <math>BRBR</math> case is the exact same by symmetry, case 2 has a probability of <math>\frac{1}{30} \cdot 2 = \frac{1}{15}</math> chance of happening. |

## Revision as of 17:13, 7 February 2020

## Problem

An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?

## Solution

Let denote that George selects a red ball and that he selects a blue one. Now, in order to get balls of each color, he needs more of both and .

There are 6 cases: (we can confirm that there are only since ). However we can clump , , and together since they are equivalent by symmetry.

and

Let's find the probability that he picks the balls in the order of .

The probability that the first ball he picks is red is .

Now there are reds and blue in the urn. The probability that he picks red again is now .

There are reds and blue now. The probability that he picks a blue is .

Finally, there are reds and blues. The probability that he picks a blue is .

So the probability that the case happens is . However, since the case is the exact same by symmetry, case 1 has a probability of chance of happening.

and

Let's find the probability that he picks the balls in the order of .

The probability that the first ball he picks is red is .

Now there are reds and blue in the urn. The probability that he picks blue is .

There are reds and blues now. The probability that he picks a red is .

Finally, there are reds and blues. The probability that he picks a blue is .

So the probability that the case happens is . However, since the case is the exact same by symmetry, case 2 has a probability of chance of happening.

and

Let's find the probability that he picks the balls in the order of .

The probability that the first ball he picks is red is .

Now there are reds and blue in the urn. The probability that he picks blue is .

There are reds and blues now. The probability that he picks a blue is .

Finally, there are reds and blues. The probability that he picks a red is .

So the probability that the case happens is . However, since the case is the exact same by symmetry, case 2 has a probability of chance of happening.