Conditional Probability

Pile A	Pile B
Pile C	Pile D

1. a card selected at random is a 7?
2. a card selected at random is from pile B?
3. a card selected from Pile D is a diamond?
4. a card selected from above that is a King was in Pile A or Pile C?

1. The probability a card selected at random is a 7 is P(7) = ³ / ₁₂



2. There are 12 possible cards that may be selected; of these, 4 are in Pile B. Therefore P(card from Pile B) = ⁴ / ₁₂
.


3. There are 5 cards in Pile D. Of these 5 cards, there are 2 diamonds. Therefore, the probability that a card is a diamond given that it is in Pile D is ² / ₅


This is an example of conditional probability. For any events (we'll call them A and E), the probability that A will occur given that event E has happened is P(A | E). (This is read "the probability of A, given E.") How do you find P(A | E)?



P(A | E) =	# of ways both A and E can occur	=	P(A and E)
	# of ways E can occur		P(E)

For this example, we are looking for the probability of getting a diamond given that we have selected a card in Pile D - this is P(diamond | card in Pile D).



P(diamond | card in Pile E) =	P(diamond and in pile D)	=	2 / 12	= 2 / 5
	________________		____
	P(in Pile D)		5 / 12

4. To find the probability that a randomly selected King came from Pile A or C we must find...
   
   

   P(card is in A or C | King) =	# of Kings in A or in C	=	(# Kings in A) + (# Kings in C)	= 3 / 4
   	________________		________________
   	# of Kings total		4