Alice and Bill are planning to have three children. (Assume that it is equally likely for a boy or a girl to be born.)
What is the probability that not all three will be of the same gender?
P(not all same)
= 1 - P(all same)
= 1 - (P(all boys) + P(all girls))
= 1 - (1/8 + 1/8)
= 3/4
To find the probability that not all three children will be of the same gender, we need to consider all the possible outcomes and count the favorable ones.
There are 2 possibilities for each child's gender: male (M) or female (F).
Let's analyze the possible outcomes:
- MMM: All three children are boys.
- FFF: All three children are girls.
- MMF, MFM, FMM: Two boys and one girl in any order.
- FFM, FMF, MFF: Two girls and one boy in any order.
Out of these 8 possible outcomes, 6 have at least one child of a different gender. Therefore, there are 6 favorable outcomes out of 8 possible outcomes.
The probability that not all three children will be of the same gender is 6/8 or 3/4.
To find the probability that not all three children will be of the same gender, we need to find the probability of three different scenarios: having two boys and one girl, having two girls and one boy, and having a boy, a girl, and another boy.
Let's break it down step by step:
1. Scenario 1: Two boys and one girl
The probability of having two boys and one girl can be calculated using the multiplication rule of probability. The probability of having a boy is 1/2, and the probability of having a girl is also 1/2. Therefore, the probability of having two boys and one girl is:
(1/2) * (1/2) * (1/2) = 1/8
2. Scenario 2: Two girls and one boy
The probability of having two girls and one boy is also 1/8, using the same logic as in Scenario 1.
3. Scenario 3: A boy, a girl, and another boy
Similarly, the probability of having a boy, a girl, and another boy is 1/8.
To find the probability that not all three children will be of the same gender, we need to add up the probabilities of the three scenarios:
1/8 + 1/8 + 1/8 = 3/8
Therefore, the probability that not all three children will be of the same gender is 3/8, or 37.5%.