To determine the theoretical probability of a family with two children having one boy and one girl, we can analyze all possible outcomes.
### Step-by-Step Solution:
1. List all possible outcomes:
When a family has two children, there are four equally likely combinations of their genders:
- Boy, Boy (BB)
- Boy, Girl (BG)
- Girl, Boy (GB)
- Girl, Girl (GG)
2. Identify favorable outcomes:
We are interested in the outcomes where the family has one boy and one girl. These outcomes are:
- Boy, Girl (BG)
- Girl, Boy (GB)
3. Count the favorable outcomes:
There are 2 favorable outcomes: BG and GB.
4. Count the total possible outcomes:
There are 4 possible outcomes in total: BB, BG, GB, GG.
5. Calculate the probability:
The theoretical probability [tex]\( P \)[/tex] of having one boy and one girl is given by the ratio of favorable outcomes to the total possible outcomes.
[tex]\[
P(\text{one boy and one girl}) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{2}{4} = \frac{1}{2}
\][/tex]
Thus, the theoretical probability of a family with two children having one boy and one girl is [tex]\(\frac{1}{2}\)[/tex] or 0.5.
### The Answer:
c. [tex]\(\frac{1}{2}\)[/tex]
