1. **Problem statement:** A married couple has two children. We want to find probabilities related to the gender of the children using tree diagrams. 2. **Assumptions:** Each child is equally likely to be a son (S) or daughter (D), with probability $\frac{1}{2}$ each. The gender of one child is independent of the other. 3. **Tree diagram outcomes:** - First child: S or D - Second child: S or D Possible outcomes: SS, SD, DS, DD, each with probability $\frac{1}{4}$. 4. **(a) Probability of having at least one son:** - Outcomes with at least one son: SS, SD, DS - Probability = $P(SS) + P(SD) + P(DS) = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4}$ 5. **(b) Probability of having no daughter:** - No daughter means both children are sons (SS) - Probability = $P(SS) = \frac{1}{4}$ 6. **(c) Probability that both are sons:** - Same as (b), $\frac{1}{4}$ 7. **(d) Possible outcomes in a tree diagram:** - First child: S or D - Second child: S or D - Outcomes: SS, SD, DS, DD Each branch splits into two with equal probability $\frac{1}{2}$, resulting in four equally likely outcomes. **Final answers:** - (a) $\frac{3}{4}$ - (b) $\frac{1}{4}$ - (c) $\frac{1}{4}$ - (d) Outcomes: SS, SD, DS, DD