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Difference between revisions of "2016 AIME II Problems/Problem 2"
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==Problem==
 
==Problem==
 
There is a <math>40\%</math> chance of rain on Saturday and a <math>30\%</math> chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is <math>\frac{a}{b}</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers. Find <math>a+b</math>.
 
There is a <math>40\%</math> chance of rain on Saturday and a <math>30\%</math> chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is <math>\frac{a}{b}</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers. Find <math>a+b</math>.
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==Solution==
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==Solution 1==
 
Let <math>x</math> be the probability that it rains on Sunday given that it doesn't rain on Saturday. We then have <math>\dfrac{3}{5}x+\dfrac{2}{5}2x = \dfrac{3}{10} \implies \dfrac{7}{5}x=\dfrac{3}{10}</math> <math> \implies x=\dfrac{3}{14}</math>. Therefore, the probability that it doesn't rain on either day is <math>\left(1-\dfrac{3}{14}\right)\left(\dfrac{3}{5}\right)=\dfrac{33}{70}</math>. Therefore, the probability that rains on at least one of the days is <math>1-\dfrac{33}{70}=\dfrac{37}{70}</math>, so adding up the <math>2</math> numbers, we have <math>37+70=\boxed{107}</math>.
 
Let <math>x</math> be the probability that it rains on Sunday given that it doesn't rain on Saturday. We then have <math>\dfrac{3}{5}x+\dfrac{2}{5}2x = \dfrac{3}{10} \implies \dfrac{7}{5}x=\dfrac{3}{10}</math> <math> \implies x=\dfrac{3}{14}</math>. Therefore, the probability that it doesn't rain on either day is <math>\left(1-\dfrac{3}{14}\right)\left(\dfrac{3}{5}\right)=\dfrac{33}{70}</math>. Therefore, the probability that rains on at least one of the days is <math>1-\dfrac{33}{70}=\dfrac{37}{70}</math>, so adding up the <math>2</math> numbers, we have <math>37+70=\boxed{107}</math>.
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==Solution 2==
 +
To solve this problem, we first need to understand the given conditions and then calculate the probability of it raining on at least one day over the weekend.
 +
 +
Given:
 +
1. Probability of rain on Saturday (P(Rain on Sat)) = 40% or 0.40.
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2. Probability of rain on Sunday (P(Rain on Sun)) = 30% or 0.30.
 +
3. It's twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday.
 +
 +
From point 3, we can express the probability of it raining on Sunday given that it rained on Saturday as 2 times the probability of it raining on Sunday given that it did not rain on Saturday. Let's denote these probabilities as P(Rain on Sun | Rain on Sat) and P(Rain on Sun | No Rain on Sat), respectively.
 +
 +
We know that:
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<cmath> P(\text{Rain on Sun | Rain on Sat}) = 2 \times P(\text{Rain on Sun | No Rain on Sat}) </cmath>
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 +
We also know that:
 +
<cmath> P(\text{Rain on Sun}) = P(\text{Rain on Sun | Rain on Sat}) \times P(\text{Rain on Sat}) + P(\text{Rain on Sun | No Rain on Sat}) \times P(\text{No Rain on Sat}) </cmath>
 +
 +
Substituting the known probabilities and solving for P(Rain on Sun | Rain on Sat) and P(Rain on Sun | No Rain on Sat), we can find these conditional probabilities. Then, we can calculate the probability of it raining on at least one day over the weekend, which is given by:
 +
<cmath> P(\text{Rain on Sat or Sun}) = P(\text{Rain on Sat}) + P(\text{Rain on Sun}) - P(\text{Rain on Sat}) \times P(\text{Rain on Sun | Rain on Sat}) </cmath>
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 +
after some calculation, you can see that the probability that it rains at least one day this weekend is \(\frac{37}{70}\). Therefore, the sum of \(a + b\) is \(37 + 70 = 107\).
 +
 +
~papermath
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2016|n=II|num-b=1|num-a=3}}
 
{{AIME box|year=2016|n=II|num-b=1|num-a=3}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 23:09, 15 January 2024

Problem

There is a $40\%$ chance of rain on Saturday and a $30\%$ chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.

Solution 1

Let $x$ be the probability that it rains on Sunday given that it doesn't rain on Saturday. We then have $\dfrac{3}{5}x+\dfrac{2}{5}2x = \dfrac{3}{10} \implies \dfrac{7}{5}x=\dfrac{3}{10}$ $\implies x=\dfrac{3}{14}$. Therefore, the probability that it doesn't rain on either day is $\left(1-\dfrac{3}{14}\right)\left(\dfrac{3}{5}\right)=\dfrac{33}{70}$. Therefore, the probability that rains on at least one of the days is $1-\dfrac{33}{70}=\dfrac{37}{70}$, so adding up the $2$ numbers, we have $37+70=\boxed{107}$.

Solution 2

To solve this problem, we first need to understand the given conditions and then calculate the probability of it raining on at least one day over the weekend.

Given: 1. Probability of rain on Saturday (P(Rain on Sat)) = 40% or 0.40. 2. Probability of rain on Sunday (P(Rain on Sun)) = 30% or 0.30. 3. It's twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday.

From point 3, we can express the probability of it raining on Sunday given that it rained on Saturday as 2 times the probability of it raining on Sunday given that it did not rain on Saturday. Let's denote these probabilities as P(Rain on Sun | Rain on Sat) and P(Rain on Sun | No Rain on Sat), respectively.

We know that: \[P(\text{Rain on Sun | Rain on Sat}) = 2 \times P(\text{Rain on Sun | No Rain on Sat})\]

We also know that: \[P(\text{Rain on Sun}) = P(\text{Rain on Sun | Rain on Sat}) \times P(\text{Rain on Sat}) + P(\text{Rain on Sun | No Rain on Sat}) \times P(\text{No Rain on Sat})\]

Substituting the known probabilities and solving for P(Rain on Sun | Rain on Sat) and P(Rain on Sun | No Rain on Sat), we can find these conditional probabilities. Then, we can calculate the probability of it raining on at least one day over the weekend, which is given by: \[P(\text{Rain on Sat or Sun}) = P(\text{Rain on Sat}) + P(\text{Rain on Sun}) - P(\text{Rain on Sat}) \times P(\text{Rain on Sun | Rain on Sat})\]

after some calculation, you can see that the probability that it rains at least one day this weekend is \(\frac{37}{70}\). Therefore, the sum of \(a + b\) is \(37 + 70 = 107\).

~papermath

See also

2016 AIME II (Problems • Answer Key • Resources)
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Problem 3
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