Integers And Probability

Number Theory Level pending

An integer from 100 through 999, inclusive, is to be chosen at random. What is the probability that the number chosen will have 0 as at least 1 digit?

171 900 \frac{171}{900} 81 900 \frac{81}{900} 271 1000 \frac{271}{1000} 19 900 \frac{19}{900} 90 900 \frac{90}{900}

Zoya Khan by Zoya Khan , 49, USA

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1 solution

Tom Engelsman
Mar 22, 2017

There are exactly 900 positive integers between 100 and 999 (inclusive). If we want those numbers containing at least one digit of zero, then we have:

100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 120, 130, 140, 150, 160, 170, 180, 190

which is 19 total between 100 and 199. This same pattern will repeat between 200 and 999 inclusively. Hence, the desired probability equals 9 ⋅ 19 900 = 171 900 . \frac{9 \cdot 19}{900} = \boxed{\frac{171}{900}}.

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