# Why $P(BB|BG \cup BB) = \frac{P(BB)}{P(BG \cup BB)}$?

I have the following question:

Determine the probability both children are boys if I have 2 children, the elder is a boy?

Then the textbook says:

Take BB, BG, GB, GG as the possible outcomes (first letter is elder kid) then: $$P(BB|BG \cup BB) = \frac{P(BB)}{P(BG \cup BB)}$$

However, how come we don't have:

$$P(BB|BG \cup BB) = \frac{P(BB \cap (BG \cup BB))}{P(BG \cup BB)}$$

which is basically what $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

I don't see how $$BB \cap (BG \cup BB)$$ would reduce to $$BB$$.

• It's helpful if you provide the name and page of the book too. – StatsStudent Feb 27 at 18:58
• @StatsStudent It's lecture notes actually. – s5s Feb 27 at 19:00
• Just make a Venn diagram of $BB$ and $BG \cup BB$ and see what ends painted by both colours – Manuel Feb 27 at 19:15

$$\begin{eqnarray*} P[BB\cap(BG\cup BB)] & = & P[(BB\cap BG)\cup(BB\cap BB)] \end{eqnarray*}$$
Now, to reduce, what do $$P(BB\cap BG)$$ and $$P(BB\cap BB)$$ reduce to?