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Difference between revisions of "1983 AHSME Problems/Problem 26"

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==Solution==
 
==Solution==
  
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Firstly note that <math>p \leq \frac{3}{4}</math> and <math>p \leq \frac{2}{3}</math>, as clearly the probability that both <math>A</math> and <math>B</math> occur cannot be more than the probability that <math>A</math> or <math>B</math> alone occurs. The more restrictive condition is <math>p \leq \frac{2}{3}</math>, since <math>\frac{2}{3} < \frac{3}{4}</math>. Now, by the Inclusion-Exclusion Principle, we also have <math>\mathrm{P}(A' \wedge B') = 1 - \mathrm{P}(A) - \mathrm{P}(B) + \mathrm{P}(A \wedge B) = 1 - \frac{3}{4} - \frac{2}{3} + p = p - \frac{5}{12}</math>, and as a probability must be non-negative, <math>p - \frac{5}{12} \geq 0</math>, so <math>p \geq \frac{5}{12}</math>. Therefore, combining our inequalities gives <math>\frac{5}{12} \leq p \leq \frac{2}{3}</math>, which is answer choice <math>\boxed{\textbf{D}}</math>.
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Firstly note that <math>p \leq \frac{3}{4}</math> and <math>p \leq \frac{2}{3}</math>, as clearly the probability that both <math>A</math> and <math>B</math> occur cannot be more than the probability that <math>A</math> or <math>B</math> alone occurs. The more restrictive condition is <math>p \leq \frac{2}{3}</math>, since <math>\frac{2}{3} < \frac{3}{4}</math>.  
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Now, by the Inclusion-Exclusion Principle, we also have <cmath>\text{P}(A' \wedge B') = 1 - \text{P}(A) - \text{P}(B) + \text{P}(A \wedge B) = 1 - \frac{3}{4} - \frac{2}{3} + p = p - \frac{5}{12},</cmath> and as a probability must be non-negative, <math>p - \frac{5}{12} \geq 0</math>, so <math>p \geq \frac{5}{12}</math>. Therefore, combining our inequalities gives <math>\frac{5}{12} \leq p \leq \frac{2}{3}</math>, which is answer choice <math>\boxed{\textbf{D}}</math>.

Revision as of 17:36, 27 January 2019

Problem

The probability that event $A$ occurs is $\frac{3}{4}$; the probability that event B occurs is $\frac{2}{3}$. Let $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval

$\textbf{(A)}\ \Big[\frac{1}{12},\frac{1}{2}\Big]\qquad \textbf{(B)}\ \Big[\frac{5}{12},\frac{1}{2}\Big]\qquad \textbf{(C)}\ \Big[\frac{1}{2},\frac{2}{3}\Big]\qquad \textbf{(D)}\ \Big[\frac{5}{12},\frac{2}{3}\Big]\qquad \textbf{(E)}\ \Big[\frac{1}{12},\frac{2}{3}\Big]$

Solution

Firstly note that $p \leq \frac{3}{4}$ and $p \leq \frac{2}{3}$, as clearly the probability that both $A$ and $B$ occur cannot be more than the probability that $A$ or $B$ alone occurs. The more restrictive condition is $p \leq \frac{2}{3}$, since $\frac{2}{3} < \frac{3}{4}$.

Now, by the Inclusion-Exclusion Principle, we also have \[\text{P}(A' \wedge B') = 1 - \text{P}(A) - \text{P}(B) + \text{P}(A \wedge B) = 1 - \frac{3}{4} - \frac{2}{3} + p = p - \frac{5}{12},\] and as a probability must be non-negative, $p - \frac{5}{12} \geq 0$, so $p \geq \frac{5}{12}$. Therefore, combining our inequalities gives $\frac{5}{12} \leq p \leq \frac{2}{3}$, which is answer choice $\boxed{\textbf{D}}$.

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