Dependent
Events


Sometimes the outcome of an event
can depend directly on the outcome
of another event.
When this happens, the events are
said to be Dependent Events
Conditional Probability
The Conditional Probability of an Event B.
P( B | A)
Is the probability that B occurs GIVEN that
A has already occurred
Ex
A Professional hockey team has 8 Wingers.
Three of these wingers are considered very
good players or snipers.
Every fall the team plays an exhibition match
with the club’s farm team. In order to make
the match more interesting for the fans, the
coaches agree to select two wingers at random
from the pro team to play for the farm team.
What is the probability that the two pro wingers
that will play for the farm team will be the
snipers?
Let A represent the event that the first winger
is a sniper
Let B represent the event that the second
winger is a sniper
Three of the Eight are picked from the pro team. So the
probability of a Pro winger being selected is
3
P( A) 
8
If the first winger selected is one of the good players, then
there are seven remaining wingers to choose from, two of
which are the good players
2
P ( B | A) 
7
The Probability that 2 snipers will be selected
is
3 2 6
3
P( AandB)   

 10.7%
8 7 56 28
Note: even though the events are
dependent upon each other you can still
multiply probabilities to find the
probability that they both happen.
Product Rule for Dependent Events
The Probability that dependent events will
occur is given by
P( AandB)  P( A)  P( B | A)
P( AandB)
P( B | A) 
P( A)
Ex
Grace’s Computer sometimes crashes when
she uses her Email Program. Her Email
program stops responding to command she
can usually close the program without the
computer crashing.
The Probability that her Email will not
respond to commands is 2.5% and the
chance that both Email and computer crash
is 1%. If Grace’s Email is failing what is the
probability that the computer will crash?
Let A be the event that Graces Email fails
Let B be the event that the Computer
Crashes
Since event A triggers event B they are
dependent upon each other. The probability
of the computer crashing depends on if the
Email fails
P( AandB)
P( B | A) 
P( A)
0.01

0.025
 0.4
 40%
Summary Note
If A and B are independent events then the
probability of both events occurring is given by
P( AandB)  P( A)  P( B)
If event B is dependent upon event A, then the
conditional probability of B given A is P(B | A).
In this case the probability of both events
P( AandB)  P( A)  P( B | A)
Homework!
Pg 334
#5,6,8,10