Question

in a family of two children, if the probabilities of a male child and a female child are both 1/2, are the events: "each child is the same sex" and "at most one male" independent?

Answer 1

P(each child is the same sex and at most one male)=0
P(at most one male)=1/2
So they are not independent

Answer 2

Independent events have no effect on the occurrence (or non-occurrence) of each other.
If one child is a male then the event "each child is the same sex" cannot occur. Therefore the events are not independent.

Answer 3

i am pretty sure this statement
P(each child is the same sex and at most one male)=0
is wrong

Answer 4

in fact i am sure it is
if there are two girls, each child is of the same sex and there is at most one male (since there is none)

Answer 5

this is not a hard sample space to write out, there are only 4 outcomes, each equally likely
(b,b), (b, g) (g, b), (g,g)

the probability of each is \(\frac{1}{4}\)

Answer 6

the probability that both children are of the same sex is \(\frac{1}{2}\) and the probability that there is at most one male \(\frac{3}{4}\)

Answer 7

put A as the event there is at most one male, B as the event the children are of the same sex, then compute
\[\frac{P(A\cap B)}{P(B)}\] and see if it is the same as \(P(A)\)

Answer 8

\[P(A)=\frac{3}{4}\]
\[P(B)=\frac{1}{2}\]
\[P(A\cap B)=\frac{1}{4}\] the last means they are both girls

Answer 9

then since
\[\frac{P(A\cap B)}{P(B)}=\frac{\frac{1}{4}}{\frac{1}{2}}=\frac{1}{2}\neq P(A)=\frac{3}{4}\] this tells you they are DEPENDENT

Answer 10

Oops sorry I've made a serious mistake there, just follow what satellite73 has said!

Answer 11

thanks everyone. do you know if it would be independent for a three child family? would i be able to find the answer by creating a new sample spacce and doing the same method?