theint481
I am confusing multiplication of 1/2 in 3 times for picking two integers.
Thus means, the probability of (Odd + Odd) = 1/2 * 1/2 = 1/4 (not 1/8).
Please kindly answer me if you get the answer. Thank you..
Check the solution below, hope it answers your question.
Two integers between 1 and 100, inclusive, each randomly and independently chosen, are either added or multiplied, with an equal chance of either operation. What is the probability that the result is even?(A) 1/3
(B) 1/2
(C) 5/8
(D) 2/3
(E) 7/8
1. The sum of two integers to be even, both of them must be even or both of them must be odd. Therefore, the probability of the sum to be even is:
P(the even sum) = P(even, even) + P(odd, odd) = 1/2*1/2 + 1/2*1/2 = 1/2.
2. The product of two integers to be even, either of them or both must be even. Therefore, the probability of the product to be even is:
P(the even product) = P(even, odd) + P(odd, even) + P(even, even) = 1/2*1/2 + 1/2*1/2 + 1/2*1/2 = 3/4.
In half of the cases we have summation and in half of the cases we have multiplication, therefore the overall probability is:
P(even result) = 1/2*1/2 + 1/2*3/4 = 5/8.
Answer: C.
Or another approach.Let's calculate the probability that the result is odd and subtract that from 1.
1. The sum of two integers to be odd, one of the integers must be odd and another even. Therefore, the probability of the sum to be odd is:
P(the odd sum) = P(even, odd) + P(odd, even) = 1/2*1/2 + 1/2*1/2 = 1/2.
2. The product of two integers to be odd, both must be odd. Therefore, the probability of the product to be odd is:
P(the odd product) = P(odd, odd) = 1/2*1/2 = 1/4.
In half of the cases we have summation and in half of the cases we have multiplication, therefore the overall probability that the result is odd is:
P(odd result) = 1/2*1/2 + 1/2*1/4 = 3/8.
Therefore, the probability that the result is even is:
P(even result) = 1 - P(odd result) = 1 - 3/8 = 5/8.
Answer: C.
Hope it's clear.