4
CONDITIONAL PROBABILITY
It is often necessary to know the probability of an event under restricted conditions.
Often when you are considering probabilities, additional information may become available that will
affect the calculation of an outcome. For example, if you are interested in the probability that Miriam
will
pass her mathematics test, then the probability of that outcome may depend on whether or not she
attended the rock concert the night before! If this information becomes available, then a more accurate
calculation of the probability can be made.
When additional information is known that will affect the probability of an outcome, we calculate
what is called a conditional probability, denoted by P(A/B), using the definition
P(AnB)
where the notation P (A/B) is read "the probability of an event A, given that the event B has occurred."
Example 1: What is the probability of rolling a sum greater than 7 with two dice if it is known that the
first die rolled is a 3?
Example 2: If a family is chosen at random from the set of all families with exactly two children, find
the
probability that:
a) the family has two boys if it is known that one child is a boy.
b) the family has two boys if it is known that the first child is a boy.
The equation for the conditional probability
may be rearranged to the form:
P(AnB) =
Example 3: The probability that Pat will go to Eastern University is 1/5. The probability that she will go
to another university is 1/2. If Pat goes to Eastern, the probability that her boyfriend Gilles will follow
her and go to Eastern is 3/4. What is the probability that both Pat and Gilles attend Eastern University?
HOMEWORK ON CONDITIONAL PROBABILITY.
1. (a) Given: P ( A n B ) = 0.5
P(B) = 0.7
Find P(A / B)
(b) Given P(S) - 0.1
P(T/S) - 0.3
FindP(SnT)
(c) Given : P (A o B ) = 0 . 2
P(A o C )
P(Bn C )
P(A)
P(B)
P(C)
= 0.3
=0.4
= 0.4
= 0.6
= 0.7
Find: (i) P (A / B)
(iv)P(C/B)
(d) Given: P ( A / B ) = 0.3
P ( B / A ) = 0.4
P ( A n B ) = 0.2
Find: (i) P(A)
(ii) P(B)
2. a) A die is rolled. What is the probability that it is a 5, given that it is greater than 3?
b) A card is drawn from a deck of cards. What is the probability that it is a Jack, given that it is a face
card.
c) In a family of four, Rob and Lesa are the parents and Michael and Jonathan are the children. Each
family member has three Christmas presents under the tree. If one present is elected at random,
what is the probability that the present is for
(i) Jonathan, if it is known that the present is for a male?
(ii) Lesa, if it is known that the present is for one of the parents?
3. A die is thrown twice. What is the probability that the sum of the rolls is less than 4 given that
(a) one of the rolls is a 1?
(b) the first roll is a one?
4. When a hockey team, the Tigers, gets possession of the puck, the following probabilities have been
estimated from team statistics: the probability they carry the puck over the opposing team's blue line
is 0.6; the probability that they get a shot on net is 0.3; the probability that they get a shot away and
score is 0.1; and the probability that they cross the blue line and score is 0.005. Find the
probability that they score given that:
(a) they cross the blue line.
(b) they get a shot away.
5. After a recent Canadian census it was discovered that among all of the families in Canada, 40 percent
have no children, 25 percent have one child, 18 percent have two children, 10 percent have three
children, 5 percent have four children, and 2 percent have five or more children. If a Canadian family
is selected at random,
(a) what is the probability that the family has more than two children?
(b) what is the probability that the family has more than two children, given that it has at least one
child?
6. Two brands of headache remedy are on the market: Acetylin and Salicin. One in 400 people taking
Acetylin suffer side effects and one in 1200 taking Salicin. At the present time it is estimated that
equal amounts of people take each kind of drug. If Acetylin is taken off the market because of
industrial sabotage during the processing of the drug, show that the probability of side effects will be
halved.