# conditional probability with independent and non independent events

I need to clear up some confusion on conditional probability and independence.

1. Two events are said to be independent if the probability of two events equal their product.

So $$P(B\mid A)=\frac{P(A \cap B)}{P(B)} = \frac{P(A)P(B)}{P(B)}=P(A),$$ correct?

Based on that i have a problem where i need to find it's conditional probability: probability of $P_c(B)$ where the event of $B=\{\text{no two people are born in the same month}\}$ and event $C= \{\text{exactly three people were born in the summer of june, july august}\}$ and there are 9 people involved.

for

$P(B)$ I got $=\frac 1{12} \frac 1{11} \frac 1{10} \frac 19 \frac18 \frac17 \frac 16 \frac 15 \frac 14 =\frac 1{79833600}$

$P(C)=\binom 9 3 \left(\frac 3{12}\right)^3\left(\frac 9{12}\right)^6=\frac{15309}{65536}$

so $P_c(B)=P(C\mid B)=\frac{P(B \cap C)}{P(C)} = \frac{P(B)P(C)}{P(C)} =\frac{(1/79833600)(15309/65536)}{15309/65536}$

This is not correct because this event is not independent. Therefore $P(C \cap B)$ does not equal $P(C)P(B)$. I can definitely see that by reading the events, it makes sense, but using the formula to check yields a different result:

$P(C\mid B)=\frac{P(C)P(B)}{P(B)} = P(C)$ then $((1/79833600) (15309/65536))/(15309/65536) = (1/79833600)$ which is $P(C)$, which says this is independent.

To be further confused my professor used $P(C\mid B)=\frac{P(C)P(B)}{P(B)} = P(B)$ to check whether it is independent or not. I thought you divide by $P(B)$ not $P(C)$. He did get the probability of $P_c(B)$ using $P_c(B) = \frac{P(B \cap C}{P(C)}$ but I thought that wasn't the correct formula if they are not independent. It seems like he did everything backwards from me which puzzles me.

If it is non independent what is the formula?

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The definition of $B$ and $C$ independent is $P(B \cap C) = P(B) P(C)$. If $P(B) \ne 0$ this is equivalent to $P(C|B) = P(C)$, and if $P(C \ne 0)$ it is equivalent to $P(B|C) = P(B)$.
If the events are not independent, you can't use $P(B) P(C)$ to calculate $P(B \cap C)$. If you happen to know $P(B|C)$, you can say $P(B \cap C) = P(B|C) P(C)$, and similarly if you happen to know $P(C|B)$ you can say $P(B \cap C) = P(C|B) P(B)$. If you don't know either of those, you have to do something else.
In your example, $B \cap C$ means exactly one person was born in June, one was born in July, and one was born in August. So the number of ways to do that with $9$ people is ...
I used this website as reference(Independence): Conditional probability and the product rule and it checks for independence by $(((P(C)*P(B))/P(B)) =P(C)$ and my professor checked by $Pc(B)=P(B)$ which is actually the opposite: $P(C|B)=(((P(C)*P(B))/P(C)) =P(B)$, assuming you can check for independence this way are both ways okay to check (I think that's what you mentioned previously). Does this mean my professor was actually looking for $P(B|C)$ instead of $P(C|B)$? – user1880760 Oct 27 '13 at 17:53