GMAT Practice Question: If x is to be chosen at random from the set \{1,2,3,4\} and y...
Question
If x is to be chosen at random from the set \{1,2,3,4\} and y is to be chosen at random from the set \{5,6,7\}, what is the probability that x y will be even?
- \frac{1}{6}
- \frac{1}{3}
- \frac{1}{2}
- \frac{2}{3}
- \frac{5}{6}
Topics: odds/evens, probability, p&c multisource
Solution
Step 1
We first express the probability of an even product in terms of the probability of an odd product.
P(\text{product is even}) = 1 - P(\text{product is odd})
Step 2
We then note that the product is odd only when both numbers are odd.
P(\text{product is odd}) = P(x \text{ is odd and } y \text{ is odd})
Step 3
We compute the probability that x is odd.
P(x \text{ is odd}) = \frac{2}{4} = \frac{1}{2}
Step 4
We compute the probability that y is odd.
P(y \text{ is odd}) = \frac{2}{3}
Step 5
We then combine these probabilities for independent choices.
P(x \text{ is odd and } y \text{ is odd}) = P(x \text{ is odd}) \times P(y \text{ is odd}) = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3}
Step 6
We then find the probability of an even product.
P(\text{product is even}) = 1 - \frac{1}{3} = \frac{2}{3}
Answer
The probability is \frac{2}{3}. (Option D)
