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I am having trouble seeing how an author reached the RHS of this conditional probability expression, particularly where the $0, 1$ coefficients came from. Could someone please explain the algebra?

$ P(C|B) = P(C|B \cap A)P(A|B) + P(C|B \cap A^C)P(A^C|B) = 1P(A|B) + 0(P(A^C | B)) $

Problem description: A bag has 1 marble, which is either green or blue, with equal probabilities. 1 green marble is added to bag (2 total marbles), and 1 green marble is removed.

Let A = event that initial marble is green, B = event that removed marble is green, C = event that remaining marble is green. Find P(C|B).

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Lewis Carroll did have a problem, that would probably have landed him in jail nowadays. But you must be thinking of another problem. It would be useful to make it clear what it is. –  André Nicolas Aug 1 '12 at 1:33
    
@AndréNicolas I don't think there is any conclusive (or any) evidence that Lewis Caroll had that problem (if it's the one I think you're talking about). –  David Mitra Aug 1 '12 at 1:39
    
A picture is worth a thousand words. In his case, there were lots of pictures. –  André Nicolas Aug 1 '12 at 1:40
    
@T. Webster, this certainly isn't right in general. For instance, by taking $A$ any event with 0 probability this would give us that every conditional probability $P(C|B)=0$. The middle of the equality is true, but the RHS you ask about only holds if $B$ and $A^C$ are mutually exclusive. –  Kevin Carlson Aug 1 '12 at 5:17
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2 Answers

up vote 1 down vote accepted

$P(C \mid (A \cap B))$ is the conditional probability that the remaining marble is green, given that the original marble was green and a green marble was drawn. It should, I hope, be obvious that $P(C \mid (A \cap B)) = 1$.

$P(C \mid (A^c \cap B))$ is the conditional probability that the remaining marble is green, given that the original marble was blue and a green marble was drawn. It should, I hope, be obvious that $P(C \mid (A^c \cap B)) = 0$.

So the rightmost expression is obtained by substituting these values in the expression in the middle. It might have been better to use $\times$ or $\cdot$ between the numbers and $P$ and thus avoid the possibility of confusion.

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This problem is used in Professor Blitzsteins' course Statistics 110 which is available from iTunes U. It is stated in Strategic Practice and Homework 2 problem 3.1. The first equality wasn't immediately clear to me so here are the details:

By the definition of conditional probability we have that $$ P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)} $$

so to restate this in terms of the problem we have that $$ P(C \mid B) = \frac{P(C \cap B)}{P(B)} $$

Now we want to find $P(C \cap B)$. By the definition of the probability function we have that the probability of the union of disjoint events are the summation of the probability of each event. If we have an event $Z$, we can decompose that into two disjoint events like this $$ P(Z) = P((Z \cap A) \cup (Z \cap A^c)) = P(Z \cap A) + P(Z \cap A^c) $$

Notice that one can also use the law of total probability to get the above result since one can always partition a sample space $S$ into $A$ and $A^c$ given $A \subseteq S$.

Let $Z = C \cap B$. This leads to $$ P(C \cap B) = P((C \cap B \cap A) \cup (C \cap B \cap A^c)) = P(C \cap B \cap A) + P(C \cap B \cap A^c) $$

By using the definition of conditional probability once on each term in the above we get $$ P(C \cap B) = P(C \mid (B \cap A)) P(A \cap B) + P(C \mid (B \cap A^c)) P(A^c \cap B) $$

Using the definition of conditional probability again on the second factor of each term we get $$ P(C \cap B) = P(C \mid (B \cap A)) P(A \mid B) P(B) + P(C \mid (B \cap A^c)) P(A^c \mid B) P(B) $$

resulting in \begin{align} P(C \mid B) &= \frac{P(C \cap B)}{P(B)} \\ &= \frac{P(C \mid (B \cap A)) P(A \mid B) P(B) + P(C \mid (B \cap A^c)) P(A^c \mid B) P(B)}{P(B)} \\ &= P(C \mid (B \cap A)) P(A \mid B) + P(C \mid (B \cap A^c)) P(A^c \mid B) \end{align}

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How does this help the OP with regards to where the '0' and '1' come from? –  Daryl Dec 29 '12 at 0:15
    
I think that has already been answered very well by Dilip Sarwate. For me the difficulty was the first part and since others might have the same difficulty I thought I would include the answer for completeness along with where the problem comes from in case another Google search ends up here. –  Henrik Schmidt Dec 29 '12 at 0:19
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