conditional probability of an event B,
in relation to event A, is the
probability that event B will occur given the knowledge that an event A
has already occurred.
|In plain English ...
You toss two pennies. The first penny shows HEADS and the
other penny rolls under the table and you cannot see it.
is the probability that they are both HEADS? Since you already
know that one is HEADS, the probability of getting HEADS on the
second penny is 1 out of 2.
The probability changes if you have partial
This "affected" probability is called
conditional probability: P(B|A)
read ... the probability of B given A.
To establish our formulas for conditional probability, we
will need to revisit our previous discussion of independent and dependent
|• If events A
and B are independent (where event A has no effect on
the probability of event B), then the conditional probability
B given event A is simply the probability of event B.
Example: Two colored dice (one blue, one
yellow) are rolled.
a. What is the probability of rolling "box cars"
b. What is the probability of rolling "box cars"
knowing the first toss is a six?
a. The probability of getting "box cars" (two
sixes) is .
If, however, we roll the dice and and see that the
blue die shows a six (and the yellow die is out of sight), the
probability of the yellow die being six is
The probability of rolling "box cars", knowing that the first
roll is a six, is
The probability changes when
you have partial information about the situation.
If events A and B are
event A has effect on the probability of event B),
then we saw that
probability that both
events occur is defined by:
P(A and B) = P(A) • P(B|A).
Dividing both sides of this equation by P(A) gives
us our formula for conditional probability
of event B given event A,
where event A affects the probability of event B:
n(A) are not zero.
|Example 2: In a school of
1200 students, 250 are seniors, 150 students take math, and 40
students are seniors and are also taking math. What is the
probability that a randomly chosen student who is a senior, is
These questions can be confusing. It sounds, at first
read, that they are asking for the probability of choosing a
student who is a senior and who is taking math. Not
It helps to re-word the question into:
the probability that the student is taking math, given that the
student is a senior.
B = the student is taking math
n(A) = the student is a senior = 250.
n(A and B) = the student is a senior and is taking math = 40.