Difference between revisions of "2004 AMC 12B Problems/Problem 4"

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The digit <math>7</math> can be either the tens digit (<math>70, 71, \dots, 79</math>: <math>10</math> possibilities), or the ones digit (<math>17, 27, \dots, 97</math>: <math>9</math> possibilities), but we counted the number <math>77</math> twice. This means that out of the <math>90</math> two-digit numbers, <math>10+9-1=18</math> have at least one digit equal to <math>7</math>. Therefore the probability is <math>\dfrac{18}{90} = \boxed{\dfrac{1}{5}} \Longrightarrow \mathrm{(B)}</math>.
 
The digit <math>7</math> can be either the tens digit (<math>70, 71, \dots, 79</math>: <math>10</math> possibilities), or the ones digit (<math>17, 27, \dots, 97</math>: <math>9</math> possibilities), but we counted the number <math>77</math> twice. This means that out of the <math>90</math> two-digit numbers, <math>10+9-1=18</math> have at least one digit equal to <math>7</math>. Therefore the probability is <math>\dfrac{18}{90} = \boxed{\dfrac{1}{5}} \Longrightarrow \mathrm{(B)}</math>.
  
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== Solution 2 ==
 
By complementary counting, we count the numbers that do not contain a <math>7</math>, then subtract from the total. There is a <math>\frac{8}{9}\cdot\frac{9}{10}</math> probability of choosing a number that does NOT contain a <math>7</math>. Subtract this from <math>1</math> and simplify yields <math>1 - \frac{8}{9}\cdot\frac{9}{10} = \frac{90}{90} - \frac{72}{90} = \frac{18}{90} = \frac{1}{5}</math>.
 
By complementary counting, we count the numbers that do not contain a <math>7</math>, then subtract from the total. There is a <math>\frac{8}{9}\cdot\frac{9}{10}</math> probability of choosing a number that does NOT contain a <math>7</math>. Subtract this from <math>1</math> and simplify yields <math>1 - \frac{8}{9}\cdot\frac{9}{10} = \frac{90}{90} - \frac{72}{90} = \frac{18}{90} = \frac{1}{5}</math>.
  

Latest revision as of 14:24, 26 October 2020

Problem

An integer $x$, with $10\leq x\leq 99$, is to be chosen. If all choices are equally likely, what is the probability that at least one digit of $x$ is a 7?

$(\mathrm {A}) \dfrac{1}{9} \qquad (\mathrm {B}) \dfrac{1}{5} \qquad (\mathrm {C}) \dfrac{19}{90} \qquad (\mathrm {D}) \dfrac{2}{9} \qquad (\mathrm {E}) \dfrac{1}{3}$

Solution

The digit $7$ can be either the tens digit ($70, 71, \dots, 79$: $10$ possibilities), or the ones digit ($17, 27, \dots, 97$: $9$ possibilities), but we counted the number $77$ twice. This means that out of the $90$ two-digit numbers, $10+9-1=18$ have at least one digit equal to $7$. Therefore the probability is $\dfrac{18}{90} = \boxed{\dfrac{1}{5}} \Longrightarrow \mathrm{(B)}$.

Solution 2

By complementary counting, we count the numbers that do not contain a $7$, then subtract from the total. There is a $\frac{8}{9}\cdot\frac{9}{10}$ probability of choosing a number that does NOT contain a $7$. Subtract this from $1$ and simplify yields $1 - \frac{8}{9}\cdot\frac{9}{10} = \frac{90}{90} - \frac{72}{90} = \frac{18}{90} = \frac{1}{5}$.

See Also

2004 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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