1. ## Help for probabilty

Can someone help me with the following problems:

1. A couple keep having children until they either have two girls or have four children. (Assume that they do not have twins, or other multiple births) and that each child they have is equally likely to be a boy or a girl independently of all other children.

i) Find the probability that they have two girls.

ii) Find the conditional probability that they have two girls given that their first child is
a boy.

iii) Find the conditional probability that their first child is a boy given that they have two
girls.

iv) Find the conditional probability that their first child is a girl given that they have two
girls.

2. Let A and B be events with P(A),P(B) > 0.
i) Show that P(Ac|B) = 1−P(A|B).

ii) Show that if P(A|B) < P(A) then P(B|A) < P(B).

iii) Illustrate both these results using question 1.

2. Here is some help on #2
i)
$\displaystyle \begin{gathered} P(B) = P(B \cap A) + P(B \cap A^c ) \hfill \\ P\left( {A^c |B} \right) = \frac{{P(B \cap A^c )}} {{P(B)}} = \frac{{P(B) - P(B \cap A)}} {{P(B)}} \hfill \\ \end{gathered}$

ii)
$\displaystyle \begin{gathered} B \cap A \subseteq A\quad \Rightarrow P(B \cap A) \leqslant P(A)\quad \hfill \\ 0 < P(B) \leqslant 1 \hfill \\ P(B)P(B \cap A) \leqslant P(B \cap A) \leqslant P(A) \hfill \\ \end{gathered}$

3. Originally Posted by maths_123
Can someone help me with the following problems:

1. A couple keep having children until they either have two girls or have four children. (Assume that they do not have twins, or other multiple births) and that each child they have is equally likely to be a boy or a girl independently of all other children.

i) Find the probability that they have two girls.

ii) Find the conditional probability that they have two girls given that their first child is
a boy.

iii) Find the conditional probability that their first child is a boy given that they have two
girls.

iv) Find the conditional probability that their first child is a girl given that they have two
girls.
[snip]
There are probably more elegant ways but I suggest you answer the questions by first listing all the possible outcomes that satisfy the restrictions and then calculate the probability for each of these outcome:

GG

BGG, GBG

BBBB,
BBBG, BBGB, BGBB, GBBB
BBGG, BGBG, GBBG

Note that conditional probability means that you have to restrict the sample space.

4. Originally Posted by mr fantastic
There are probably more elegant ways but I suggest you answer the questions by first listing all the possible outcomes that satisfy the restrictions and then calculate the probability for each of these outcome:

GG

BGG, GBG

BBBB,
BBBG, BBGB, BGBB, GBBB
BBGG, BGBG, GBBG

Note that conditional probability means that you have to restrict the sample space.
Thanks for the help.