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Difference between revisions of "2005 iTest Problems/Problem 10"

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==Problem==
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The probability of U2 dismantling an atomic bomb is <math>11\%</math>. The probability of Coldplay finding X & Y is <math>23\%</math>. If the probability of both events occurring is <math> 6\%,</math> find the probability that neither occurs.
 
==Solution 1==
 
==Solution 1==
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Let <math>x = \frac{43 - 22y}{21}</math> such that <math>1 < \frac{43 - 22y}{21} < 11</math>. This means that <math>-\frac{94}{11} < y < 1</math>. If <math>y \in \mathbb{Z}</math>, then <math>y ={-8,-7,-6,-5,-4,-3,-2,-1,0}</math>. However, none of these values for <math>y</math> results in a complementary integral value for <math>x</math>. Therefore, there are <math>\boxed{0}</math> integer solutions <math>x, y \in \mathbb{Z}</math> that solves <math>21x + 22y = 43</math> over <math>1 < x < 11</math> and <math>y < 22</math>.
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By the Principle of Inclusion-Exclusion, the probability of U2 dismantling an atomic bomb and Coldplay NOT finding X & Y is <math>11\%-6\%=5\%</math>. Similarly, the probability of Coldplay finding X & Y and U2 NOT dismantling an atomic bomb is <math>17\%</math>. Therefore, the probability that neither occurs is <math>100\%-5\%-17\%-6\%=\boxed{72\%}</math>.
  
 
==See Also==
 
==See Also==
 
{{iTest box|year=2005|num-b=9|num-a=11}}
 
{{iTest box|year=2005|num-b=9|num-a=11}}
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[[Category: Introductory Combinatorics Problems]]

Latest revision as of 19:02, 13 October 2025

Problem

The probability of U2 dismantling an atomic bomb is $11\%$. The probability of Coldplay finding X & Y is $23\%$. If the probability of both events occurring is $6\%,$ find the probability that neither occurs.

Solution 1

By the Principle of Inclusion-Exclusion, the probability of U2 dismantling an atomic bomb and Coldplay NOT finding X & Y is $11\%-6\%=5\%$. Similarly, the probability of Coldplay finding X & Y and U2 NOT dismantling an atomic bomb is $17\%$. Therefore, the probability that neither occurs is $100\%-5\%-17\%-6\%=\boxed{72\%}$.

See Also

2005 iTest (Problems, Answer Key)
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