To find the probability of a couple having at least two boys out of four children, we first consider the total possible outcomes of having four children. Each child can either be a boy (B) or a girl (G). Thus, the total number of outcomes for four children can be calculated using the formula:
Total Outcomes = 2n
Where n is the number of children. In this case, n = 4, so:
Total Outcomes = 24 = 16
Next, we will determine the different combinations of having at least two boys. To simplify calculations, we can use the complement rule, which involves calculating the probability of the opposite event—having fewer than two boys (which means either 0 boys or 1 boy)—and then subtracting that probability from 1.
1. **Calculating the probability of having 0 boys (all girls)**:
- Combination: GGGG (1 way)
2. **Calculating the probability of having 1 boy**:
- The child can be a boy in one of the 4 positions, with the other 3 children being girls. The combinations are:
- BGGG
- GBGG
- GGBG
- GGGB
So, there are 4 ways to have exactly 1 boy.
Now, we can sum the total combinations for fewer than two boys:
- Ways to have 0 boys: 1
- Ways to have 1 boy: 4
- Total ways to have less than 2 boys = 1 + 4 = 5
Next, we calculate the probability of getting fewer than two boys:
- Probability (fewer than 2 boys) = Number of favorable outcomes / Total outcomes = 5 / 16
Finally, we find the probability of having at least 2 boys:
- Probability (at least 2 boys) = 1 – Probability (fewer than 2 boys)
- Probability (at least 2 boys) = 1 – 5/16 = 16/16 – 5/16 = 11/16
Thus, the probability that the couple will have at least two boys when they have four children is:
11/16 or approximately 68.75%