Difference between revisions of "2008 AMC 10B Problems/Problem 17"

Latest revision as of 12:42, 30 January 2024

Problem

A poll shows that $70\%$ of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work?

$\mathrm{(A)}\ {{{0.063}}} \qquad \mathrm{(B)}\ {{{0.189}}} \qquad \mathrm{(C)}\ {{{0.233}}} \qquad \mathrm{(D)}\ {{{0.333}}} \qquad \mathrm{(E)}\ {{{0.441}}}$

Solution 1

Letting Y stand for a voter who approved of the work, and N stand for a person who didn't approve of the work, the pollster could select responses in $3$ different ways: $\text{YNN, NYN, and NNY}$ . The probability of each of these is $(0.7)(0.3)^2=0.063$ . Thus, the answer is $3\cdot0.063=\boxed{\mathrm{(B)}\ {{{0.189}}}}$

Solution 2

In more concise terms, this problem is an extension of the binomial distribution. We find the number of ways only 1 person approves of the mayor multiplied by the probability 1 person approves and 2 people disapprove: ${3\choose 1} \cdot(0.7)^1\cdot(1-0.7)^{(3-1)}=3\cdot0.7\cdot0.09=\boxed{\mathrm{(B)}\ {{{0.189}}}}$

Video Solution by TheBeautyofMath

With explanation of how it helps on future problems, emphasizing "Don't Memorize, Understand" https://youtu.be/PO3XZaSchJc

~IceMatrix

2008 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 4: / Line 4: @@
 <math>\mathrm{(A)}\ {{{0.063}}} \qquad \mathrm{(B)}\ {{{0.189}}} \qquad \mathrm{(C)}\ {{{0.233}}} \qquad \mathrm{(D)}\ {{{0.333}}} \qquad \mathrm{(E)}\ {{{0.441}}}</math>
-==Solution==
+==Solution 1==
-The pollster could select responses in 3 different ways: YNN, NYN, and NNY, where Y stands for a voter who approved of the work, and N stands for a person who didn't approve of the work. The probability of each of these is <math>(0.7)(0.3)^2=0.063.</math> Thus, the answer is <math>3 \cdot 0.063=0.189\Rightarrow \boxed{B}</math>
+Letting Y stand for a voter who approved of the work, and N stand for a person who didn't approve of the work, the pollster could select responses in <math>3</math> different ways: <math>\text{YNN, NYN, and NNY}</math>. The probability of each of these is <math>(0.7)(0.3)^2=0.063</math>. Thus, the answer is <math>3\cdot0.063=\boxed{\mathrm{(B)}\ {{{0.189}}}}</math>
-==Alternative Solution==
-In more concise terms, this problem is an extension of the binomial distribution. We find the number of ways only 1 person approves of the mayor multiplied by the probability 1 person approves and 2 people disapprove: <cmath>{3\choose 1} \cdot (0.7)^1\cdot(1-0.7)^{(3-1)} = 3 \cdot 0.7 \cdot 0.09 = 0.189 \Rightarrow \boxed{B}</cmath>
+==Solution 2==
+In more concise terms, this problem is an extension of the binomial distribution. We find the number of ways only 1 person approves of the mayor multiplied by the probability 1 person approves and 2 people disapprove: <math>{3\choose 1} \cdot(0.7)^1\cdot(1-0.7)^{(3-1)}=3\cdot0.7\cdot0.09=\boxed{\mathrm{(B)}\ {{{0.189}}}}</math>
 ==Video Solution by TheBeautyofMath==

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