4-CONDITIONAL PROBABILITIES
Suppose that we toss 2 dice, and suppose that each of the 36 possible outcomes
is equally likely to occur and hence has probability 
Suppose further that we observe that the first die is a 3. Then, given this information,
what is the probability that the sum of the 2 dice equals 8?
To calculate this probability, we reason as follows: Given that the initial die is a 3,
there can be at most 6 possible outcomes of our experiment,
namely, (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), and (3, 6).
Since each of these outcomes originally had the same probability
of occurring, the outcomes should still have equal probabilities.
That is, given that the first die is a 3, the (conditional) probability of
each of the outcomes (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), and (3, 6) is 
, whereas
the (conditional) probability of the other 30 points in the sample space is 0.
Hence, the desired probability will be
If we let E and F denote, respectively, the event that the sum of the dice is 8 and the event
that the first die is a 3, then the probability just obtained is called
the conditional probability that E occurs given that F has occurred and is denoted by
P(E|F)
If P(F) > 0, then
P(E|F)= 
A coin is flipped twice. Assuming that all four points in the sample
space S = {(h, h), (h, t), (t, h), (t, t)} are equally likely, what is
the conditional probability that both flips land on heads, given that
(a) the first flip lands on heads?
(b) at least one flip lands on heads?
Let B = {(h, h)} be the event that both flips land on heads;
let F = {(h, h), (h, t)} be the event that the first flip lands on heads; and
let A = {(h, h), (h, t), (t, h)} be the event that at least one flip lands on heads.
The probability for (a) can be obtained from
Thus, the conditional probability that both flips land on heads
given that the first one does is 
, whereas the conditional probability
that both flips land on heads given that at least one does is only 
Many students initially find this latter result surprising.
They reason that, given that at least one flip lands on heads,
there are two possible results: Either they both land on heads or only one does.
Their mistake, however, is in assuming that these two possibilities are equally likely.
For, initially, there are 4 equally likely outcomes.
Because the information that at least one flip lands on heads is equivalent to the information
that the outcome is not (t, t), we are left with the 3 equally
likely outcomes (h, h), (h, t), (t, h), only one of which results in both flips landing on heads.
In the card game bridge, the 52 cards are dealt out
equally to 4 players—called East, West, North, and South.
If North and South have a total of 8 spades among them, what is the probability
that East has 3 of the remaining 5 spades?
Probably the easiest way to compute the desired probability is to work
with the reduced sample space. That is, given that North–South have a total
of 8 spades among their 26 cards, there remains a total of 26 cards,
exactly 5 of them being spades, to be distributed among the East–West hands.
Since each distribution is equally likely, it follows that the conditional
probability that East will have exactly 3 spades among his or her 13 cards is
An experiment is conducted to examine the relationship between
cigarette smoking and cancer. The individuals are randomly selected from
the male population of a certain section of the United States.
The results are summarized as follows:
Find P(developing cancer|smoker)
Call a household prosperous if its income exceeds $100,000.
Call the household educated if the household completed college.
Select an American household at random, and let A be the event that the selected
household is prosperous and let B be the event that it is educated.
According to the Current Population Survey, P(A) = 0.134, P(B) = 0.254, P(A|B) = 0.080.
(a) Find the conditional probability that a household is educated, given that it is prosperous.
(b) Find the conditional probability that a household is prosperous, given that it is educated.
The multiplicative rule:
by P(F ), we obtain
𝑃(𝐸∩𝐹)=𝑃(𝐹)P(𝐸|𝐹).
In words, this Equation states that the probability that both E and F occur
is equal to the probability that F occurs multiplied by the
conditional probability of E given that F occurred. This Equation is
often quite useful in computing the probability of the intersection of events.
Celine is undecided as to whether to take a French course or a chemistry course.
She estimates that her probability of receiving an A grade would be 
in a French
course and 
in a chemistry course. If Celine decides to base her decision on the flip
of a fair coin, what is the probability that she gets an A in chemistry?
Let the event that Celine takes chemistry C and A denote the event that she receives an A
in whatever course she takes, then the desired probability is P(C 
A), which is calculated as follows:
A focus group of 10 consumers has been selected to view a new TV commercial.
After the viewing, 2 members of the focus group will be randomly selected and asked
to answer detailed questions about the commercial. The group contain 4 men and 6 women.
What is the probability that the 2 chosen to answer questions will both be women?
P(first person is female ) = P(second person is
female | first person is female ) = P(both people are female) =
A box has three tickets, colored red, white and blue.
Two tickets will be drawn at random without replacement.
What is the chance of drawing the red and then the white?
If we randomly pick two television tubes in succession from a shipment
of 240 television tubes of which 15 are defective, what is the probability
that they will both be defective? A generalization of Equation 𝑃(𝐸∩𝐹)=𝑃(𝐹)P(E|F),
which provides an expression for the probability of the intersection of an arbitrary
number of events, is sometimes referred to as the multiplication rule.
To prove the multiplication rule, just apply the definition of
conditional probability to its right-hand side, giving
