IB MATH STUDIES
Topic 3 Day 5 – Conditional Probability
page 1
PROBABILITY & VENN DIAGRAMS:
EVENT
SET LANGUAGE
The complement of A,
denoted A'.
The intersection of A
and B: A B
The union of events A
and B: AB
If A B =  the events
A and B are said to be
disjoint. That is, they
have no elements in
common.
ex.
ex.
VENN DIAGRAM
A' is the complement to
the set A. That is, the
set of elements that do
not belong to set A.
A B is the intersection
of the sets A and B.
That is, the set of
elements belonging to
both the set A and the
set B.
AB is the union of the
sets A and B, that is the
set of elements
belonging to A or B or
both A and B.
PROBABILITY RESULT
U
A'
A
U
A
p(A') = 1 - p(A)
p(A') is the probability
that A does not occur.
p(A B) is the probability
that both A and B occur.
B
p(AB) is the probability
that either event A or
event B (or both) occur.
B
From this we have what
is known as the
'ADDITION RULE' for
probability:
p(AB) = p(A) + p(B) - p(A B)
U
A
If A B =  the sets A
and B are said to be
mutually exclusive.
U
A
B
If A and B are mutually
exclusive events then
event A and event B
cannot occur
simultaneously, that is:
n(AB) = 0
p(AB) = 0
Thus,
p(AB) = p(A) + p(B)
If p(A) = 0.35, p(B) = 0.5 and p(AB) = 0.15. Find:
(i)
p(A')
(ii)
p(AB)
(iii)
p(AB')
A bag has 20 coins numbered from 1 to 20. A coin is drawn at random and its number noted.
What is the probability that the coin has a number that is divisible by 3 or by 5?
IB MATH STUDIES
Topic 3 Day 5 – Conditional Probability
page 2
CONDITIONAL PROBABILITY: Conditional probability works in the same way as simple probability. The only
difference is that we are provided with some prior knowledge (or some extra condition about the outcome). So rather
than considering the whole sample space,
 , given some extra information about the outcome of the experiment, we
only need to consider part of the sample space, or
ex.
* .
A die is tossed.
(a)
What is the probability that it is a 2?
(b)
After tossing a die, it is noted that an even number appeared. What is the probability that it is a two?
FORMAL DEFINITION OF CONDITIONAL PROBABILITY:
If A and B are two events, then the conditional probability of event A given event B is found using:
p(A|B) =
Note:
1.
2.
3.
p(A  B)
, given that p(B)  0.
p(B)
If A and B are mutually exclusive, the p(A|B) = 0
From the above rule, we also have the general Multiplication Rule:
p(A  B) = p(A|B) x p(B)
Generally: p(A|B)  p(B|A)
ex.
Two dice, numbered one to six, are rolled onto a table. Find the probability of obtaining a sum of five given
that the sum is seven or less.
ex.
Two events A and B are such that p(A) = 0.5, p(B) = 0.3, and p(A  B) = 0.6
(a)
Find p(A|B)
(b)
Find p(B|A)
(c)
Find p(A'|B)
IB MATH STUDIES
Topic 3 Day 5 – Conditional Probability
INDEPENDENCE:
page 3
Two events are said to be independent if the probability of event B occurring is not influenced
by event A occuring.
Two events A and B are independent if, and only if,
p(A|B) = p(A) and p(B|A) = p(B)
A and B are independent if, and only if
p(AB) = p(A) x p(B)
This definition can be used to decide if two events are independent. However, as a rule of thumb, if two events are
"physically independent" then they will also be statistically independent.
Note:
1.
Never assume that two events are independent unless you are absolutely certain that they are
independent
2.
How can you tell if two events are independent? If they are physically independent, they are
mathematically independent.
3.
Make sure that you understand the difference between mutually exclusive events and independent
events.
Mutually exclusive means that the events A and B have nothing in common and so there is no
intersection. That is AB =  therefore p(AB) = 0
Independent means that the outcome of event A will not influence the outcome of event B. That is,
p(AB) = p(A) x p(B)
4.
Independence can be extended to more than two events. If events A, B, and C are independent, then
p(ABC) = p(A) x p(B) x p(C)
5.
When do I add the probabilities and when do I multiply? If you are asked to determine the
probability of one event or the other event occurring, you add their individual probabilities. If you are
asked to determine the probability of one event and the other event occurring, you multiply their
individual probabilities. Again, you are assuming that the events are independent.
ex.
Two fair dice are rolled. Find the probability that two even numbers will show up.
ex.
Debra has a 70% chance of winning the 100m race and a 60% chance of winning the 200m race.
(a)
Find the probability that she only wins one race.
(b)
Find the probability that she wins both races.
IB MATH STUDIES
Topic 3 Day 5 – Conditional Probability
page 4
WITH REPLACEMENT vs. WITHOUT REPLACEMENT: The probability of an event a subsequent event occurring,
for example picking a certain colored marble out of a bag, depends upon whether the first choice is returned to the
sample space for re-selection.
If the choice is returned for re-selection (WITH REPLACEMENT) then the probability determination maintains a fixed
number of elements in the sample space.
If the choice is not returned for re-selection (WITHOUT REPLACEMENT) then the probability determination is based
upon the sample space being reduced by one item and the event has one less possible occurrence.
This can best be seen in an example:
ex.
ex.
A box contains 2 red cubes and 4 black cubes. If two cubes are selected at random, find the probability that
both cubes are red given that:
(a)
the first cube is replaced before the next cube is selected.
(b)
the first cube is not replaced before the second cube is selected.
A bag contains 5 white balls and 4 red balls. Two balls are selected in such a way that the first ball drawn is
not replaced before the next ball is drawn. Find the probability of selecting exactly one white ball.