Probability & Statistics
Section 6.2
“Conditional Probability and the Multiplication Rule”

A conditional probability is the probability of an event occurring, GIVEN that another event has already
occurred.
The conditional probability of Event B occurring, given that Event A has already occurred, is denoted by
P( B A) and is read as “the probability of B given A.”
(example 1)
Two cards are selected in sequence from a standard deck of 52 cards. Find the probability that the second
card is a Queen, given that the first card is a King.
P(Queen King ) 
4
51
**The condition sets the value in the denominator.
Since one card (the King) has already been drawn,
there are now only 51 cards left to draw from.
The numerator is 4, since there are 4 Queens in the
deck.
(example 2)
The results of a particular study show a child’s IQ and the presence of a particular gene. What is the
probability that a child has a high IQ, given that the child has the gene?
High IQ
Normal IQ
TOTAL
Gene Present
33
39
72
P( HighIQ Gene Pr esent ) 
Gene NOT present
19
11
30
TOTAL
52
50
102
33
72
The denominator is 72 because the condition (the “given”) is the gene being present, so those are the
only children we’re talking about. Out of those 72 kids, 33 of them had a high IQ.
1

Independent events do NOT affect the outcomes of one another. For example – if you flip a coin and
then roll a die, those are independent events. The result you get from flipping the coin has absolutely
no influence whatsoever on what number you get when you roll the die.
For independent events, the probability of the 2nd event and its conditional probability given that the
1st event already occurred are the same:

P( B)  P( B A)
The Multiplication Rule for joint events:
P(A and B) = P( A)  P( B A)
OR
If A and B are independent, then P(A and B) = P( A)  P( B)
EXAMPLE:
For anterior cruciate ligament (ACL) reconstructive surgery, the probability that the surgery is successful is
0.95 (or 95%).
(a) Find the probability that three ACL surgeries are all successful.
P(success AND success AND success)
(0.95)(0.95)(0.95)
0.8574
85.74%
(b) Find the probability that NONE of the three surgeries is successful.
P(failure AND failure AND failure)
(0.05)(0.05)(0.05)
0.000125
0.0125%
(c) Find the probability that AT LEAST ONE of the three surgeries is successful.
1 – P(NONE are successful)
1 – 0.000125
0.999875 or 0.9999 if you round
99.99%
2
EXAMPLE:
About 16,500 U.S. medical school seniors applied to residency programs in 2012. Ninety-five percent of the
seniors were matched with residency positions. Of those, 81.6% were matched with one of their top three
choices. Medical students rank the residency programs in their order of preference, and program directors in
the U.S. rank the students. The term “match” refers to the process whereby a student’s preference list and
program director’s preference list overlap, resulting in the placement of the student in a residency program.
A Venn Diagram to illustrate the setting:
(a) Find the probability that a randomly selected senior was matched with a residency position AND it was
one of their top three choices.
P(matched with a position AND it was one of their top three choices)
(0.95)(0.816)
0.7752
77.52%
(b) Find the probability that a randomly selected senior who was matched with a residency position did
NOT get matched with one of their top three choices.
P(DidNotGet1ofTop3 MatchedWithA Re sidencyPosition)
Since these are COMPLEMENTARY events (one is where it DOES happen, the other is where it
does NOT), …
1  P( DidGet1ofTop3 MatchedWithA Re sidencyPosition)
1 – 0.816
0.184
18.4%
ASSIGNMENT:
Worksheet Packet 6.2
(7 – 15, 17, 19, 20, 22, 23, 25, 27)
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