~This is one of several problem types most missed by beginning students. When
first exposed to this type, most students find it difficult to understand the basic
reasoning behind getting the correct answer. Before I formalize it with the symbols used, let me give you an easy way to understand
example.
~In a standard deck of playing cards, there are 4 suits, clubs,
diamonds, hearts, and spades. There are 13 of each. So, if you wanted
to calculate the probability of pulling
a heart, the answer would be 13/52 or 1/4 reduced.
~However, someone else pulls a card and looks at it but does not tell
you what it is. That person puts it on a table, face down and asks you
to give the probability of that card being a heart. Normally, you would
say, 13/52. But, this time, that person gives you a hint since he or
she has seen it. "It's a red card", is told to you. Well, now you know
that the card which is face down in front of you is either a diamond or
a heart. There are 26 possibilities now, not 52. All 26 black cards
have been eliminated from possible outcomes. What this means, is that
the sample space has changed from 52 to 26. So, your probability has
changed from 13/52 to 13/26. Your chances that a heart is on the
table has just increased to 1/2 or 50% (because of the hint given).
~This example displays what is at the core of conditional
probability. The sample space has changed due to given
information. That given information (the hint in the above example) is
the fact that an event has already occurred (a red card has been
chosen). So, now, the sample space has changed to just red cards. So,
your probability calculation is now based on this new sample space.
~In symbols, it looks like this: P(A|B) means "the probability of
event A, given that event B has already occurred". So, the new sample
space is that of B.
~It usually takes beginning students a little time for this concept to "sink in".
~Another way to visualize this is by the use of a Venn diagram. Sketch
a rectangle representing the entire sample space of the original
experiment. Then sketch two overlapping circles inside this rectangle.
One representing event A and the other representing event B. Say, event
B has occurred and you wanted to calculate the probability of event A
under that circumstance. Well, you must use the area enclosed by circle
B and look for any part of the circle representing event A that is in
circle B. This is the same as saying, "take all points in A that are in
B" for your calculation of P(A|B), In our example, we take the
number of hearts (event A) that are in the set of all red cards (event
B).
~Hope your understanding of this tricky probability is enhanced.