2004 AMC 12B Problems/Problem 4

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 $\dfrac{18}{90} = \boxed{\dfrac{1}{5}} \Longrightarrow \mathrm{(B)}$ .

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 $1 - \frac{8}{9}\cdot\frac{9}{10} = \frac{90}{90} - \frac{72}{90} = \frac{18}{90} = \frac{1}{5}$ .

2004 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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2004 AMC 12B Problems/Problem 4

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