If it is known that the family has at least one boy, what is the probability that the oldest child is a boy?
My attempt at this: The probability that the known boy, call it $boy_0$ is the oldest is:
$p(boy_0)= \frac{1}{5}$
Among the other children the probability of being a boy and the oldest is:
$p(boy_{old}): \frac{1}{2} \cdot \frac{1}{5} = \frac{1}{10}$
Now, the probability of being a boy and the oldest for the known and unknown sets of children is
$p(boy_0)+p(boy_{old})\frac{1}{5} + \frac{1}{10} =\frac{3}{10}$
Am I correct here?
What is the probability that there are 2 boys and 3 girls?
My attempt:
Probability that there are 2 boys out of 5 children is:
$p(boy) = {5\choose 2} (0.5)^2 (1-0.5)^{5-2}= \frac{5}{16} $
Probability that there are 3 girls out of 5 children is:
$p(girl)= {5\choose 3} (0.5)^3 (1-0.5)^{5-3} = \frac{5}{16}$
Probability that there are 2 boys and 3 girls is: $p(boy and girl) = p(boy) \cdot p(girl) = \frac{5}{16} \cdot \frac {5}{16} = \frac{25}{256}$ Correct or not?
Lastly,
In this family of 2 boys and 3 girls, what is the probability that the oldest child is a boy?
Probability of being oldest is $\frac{1}{5}$ and probability of being a boy is$ \frac{2}{5}$. Thus the probability of the oldest being a boy is $\frac{1}{5} \cdot \frac{2}{5} = \frac{2}{10}.$
I'm a bit shakey about all my answers. Are any of them correct?