# Thread: Need help on some problems

1. ## Need help on some problems

If P(A/B)=P(A/B') then show that A and B are independent?

2. ## Re: Need help on some problems

given $P[A|B] = P[A | B']$

Note that clearly $B \cup B' = U$ and $B \cap B' = \emptyset$ so $P[B]+P[B']=1$

\begin{align*} &P[A] = \\ \\ &P[A|B]P[B] + P[A|B']P[B'] = \\ \\ &P[A|B]P[B] + P[A|B]P[B'] = \\ \\ &P[A|B](P[B]+P[B'])= \\ \\ &P[A|B] \end{align*}

multiply both sides by $P[B]$

$P[A]P[B] = P[A|B]P[B] = P[A \wedge B]$

and thus $A$ and $B$ are independent.

3. ## Re: Need help on some problems

Originally Posted by lera
If P(A/B)=P(A/B') then show that A and B are independent?
$\begin{gathered} P(A|B) = P(A|B') \hfill \\ \frac{{P(A \cap B)}}{{P(B)}} = \frac{{P(A \cap B')}}{{P(B')}} \hfill \\ \frac{{P(A \cap B)}}{{P(B)}} = \frac{{P(A \cap B')}}{{1 - P(B)}} \hfill \\ [1 - P(B)]P(A \cap B) = P(B)P(A \cap B') \hfill \\ P(A \cap B) = P(B)[P(A \cap B) + P(A \cap B')] \hfill \\ P(A \cap B) = P(B)P(A) \hfill \\ \hfill \\ \end{gathered}$

4. ## Re: Need help on some problems

Originally Posted by romsek
given $P[A|B] = P[A | B']$

Note that clearly $B \cup B' = U$ and $B \cap B' = \emptyset$ so $P[B]+P[B']=1$

\begin{align*} &P[A] = \\ \\ &P[A|B]P[B] + P[A|B']P[B'] = \\ \\ &P[A|B]P[B] + P[A|B]P[B'] = \\ \\ &P[A|B](P[B]+P[B'])= \\ \\ &P[A|B] \end{align*}

multiply both sides by $P[B]$

$P[A]P[B] = P[A|B]P[B] = P[A \wedge B]$

and thus $A$ and $B$ are independent.
Thanks !!!

5. ## Re: Need help on some problems

Originally Posted by Plato
$\begin{gathered} P(A|B) = P(A|B') \hfill \\ \frac{{P(A \cap B)}}{{P(B)}} = \frac{{P(A \cap B')}}{{P(B')}} \hfill \\ \frac{{P(A \cap B)}}{{P(B)}} = \frac{{P(A \cap B')}}{{1 - P(B)}} \hfill \\ [1 - P(B)]P(A \cap B) = P(B)P(A \cap B') \hfill \\ P(A \cap B) = P(B)[P(A \cap B) + P(A \cap B')] \hfill \\ P(A \cap B) = P(B)P(A) \hfill \\ \hfill \\ \end{gathered}$
Thanks !!