swerve wrote: ↑

Fri Nov 13, 2020 10:20 am

If a six sided die is rolled three times, what is the probability of getting at least one even number and at least one odd number?

A. 1/8
B. 1/4
C. 1/2
D. 3/4
E. 7/8

The OA is D

Solution:

The phrases “at least one even number” and “at least one odd number” translate as “two even numbers and one odd number” or “one even number and two odd numbers,”. The probability of getting the former is 1/2 x 1/2 x 1/2 x 3C2 = 1/8 x 3 = 3/8 (notice that 3C2 = 3 is the number of ways of arranging two even numbers and one odd number). Similarly, the probability of getting the latter is also 3/8. Therefore, the overall probability is 3/8 + 3/8 = 6/8 = 3/4.

Alternate Solution:

When a six sided die is rolled three times, the total number of possible outcomes is 6 x 6 x 6 = 216.

Notice that if “getting at least one even number and at least one odd number” is not satisfied, then the outcome consists of either all even or all odd numbers. Of the 216 possible outcomes, 3 x 3 x 3 = 27 of them consist of all even numbers and by the same reasoning, another 27 of them consist of all odd numbers. Thus, 27 + 27 = 54 of the outcomes do not meet the requirement of containing at least one even number and at least one odd number. It follows that 216 - 54 = 162 of the outcomes meet this requirement and hence, the probability is 162/216 = 3/4.

Answer: D