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 $BB$.

  • $\begingroup$ It's helpful if you provide the name and page of the book too. $\endgroup$ – StatsStudent Feb 27 at 18:58
  • $\begingroup$ @StatsStudent It's lecture notes actually. $\endgroup$ – s5s Feb 27 at 19:00
  • 1
    $\begingroup$ Just make a Venn diagram of $BB$ and $BG \cup BB$ and see what ends painted by both colours $\endgroup$ – Manuel Feb 27 at 19:15

You both are right. But here's a hint to get the answer in your notes: According to Probability Distributive Laws:

\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?


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