Official Guide Explanation:
Problem Solving #79

Background

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Solution and Metadata

Question: 79
Page: 72
Difficulty: 6 (Moderately Difficult)
Category 1: Arithmetic > Properties of Integers > Evens and Odds
Category 2: Arithmetic > Discrete Probability >

Explanation: When two integers are multiplied together, the result is even if one or both of the integers is even. In this case, we want the probability that the result is even; rather than finding the probability that one both is even, it's easier to find the probability that the result is odd. For the result to be odd, both integers must be odd. The probability of two indepedent events occuring is (p₁)(p₂), so the probability of the result being odd is the probability of the first integer being odd times the probability of the second integer being odd.

Since two of the four numbers in the first set are odd, the probability of the first integer being odd is (2/4) = (1/2). The second set has two odd numbers out of a total of three, so the probability that the second number is odd is (2/3). The probability of the result being odd, then, is ((1/2))((2/3)) = (1/3). However, we're interested in the probability that the result is even, so we have to subtract the probability that it's odd from 1: 1 - (1/3) = (2/3), choice (D).

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