MATH 141-501
Section 7.5 Lecture Notes
Conditional Probability
Sometimes, knowing that an event has already occurred changes our perception of the likelihood (i.e. the probability) that another event will occur.
Example: A pair of fair dice is rolled.
Question 1: What is the probability that the sum of the numbers showing
is greater than 10?
Question 2: What is the probability that the sum of the numbers showing
is greater than 10, given that one of the numbers is a 6?
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Conditional Probability
Definition: If A and B are events in a sample space S, the conditional
probability of B given A, denoted P (B | A), is the probability that B will
occur, given that we already know A has occurred.
Formula for Conditional Probability: If A and B are events in an
experiment and P (A) 6= 0, then the conditional probability that the event B
will occur, given that the event B has already occurred is
P (B | A) =
2
P (A ∩ B)
.
P (A)
Example: Let A and B be two events in a sample space S such that
P (A) = .61, P (B) = .49, and P (A ∩ B) = .2.
Find...
...P (A | B)
...P (B | A)
...P (A | B c )
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Example (from textbook):
Receiving
Financial
Aid
Undergraduates
Graduates
Total
4,222
1,879
6,101
Not Receiving
Financial
Aid
3,898
731
4,629
Total
8,120
2,610
10,730
Let A be the event that a student selected at random, and let B be the event
that a student selected at random is receiving financial aid.
Find the following probabilities...
...P (A)
...P (B)
...P (A ∩ B)
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Example (continued) :
Receiving
Financial
Aid
Undergraduates
Graduates
Total
4,222
1,879
6,101
Not Receiving
Financial
Aid
3,898
731
4,629
Total
8,120
2,610
10,730
What are the following probabilities?
... P (B | A)
... P (B | Ac )
Question: Are the events A and B independent events?
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Product Rule
If take the formula P (B | A) =
we obtain the...
P (A∩B)
P (A)
and multiply both sides by P (A),
Product Rule: P (A ∩ B) = P (A) · P (B | A)
In Words: The probability of A AND B is the probability of A
times the probability of B given A.
Multi-stage experiments and Tree diagrams
A multi-stage experiment is an experiment which consists of several
parts. The outcomes and probabilities at each stage could depend on what
happened in previous stages. Tree diagrams (which we have seen before) are
useful for solving problems related to these multi-stage experiments.
Procedure for Drawing a Tree Diagram: I tried to think of a clear
step-by-step description for how to create a tree diagram, but I couldn’t come
up with anything which was brief and clear. Let’s do an example, and see if we
can come up with a recipe for how to use these important tools.
Example: (from textbook) An experiment consists of drawing two cards
from a well-shuffled deck. What is the probability that the second card is a
club?
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Example (from textbook)
A box contains seven batteries. Two of these batteries are defective. The
batteries are taken out of the box randomly and tested until a nondefective one
is found. (We do not put a battery back in the box after we test it.)
Question: What is the maximum number of batteries that could be tested?
Tree Diagram:
Question: What is the probability that the number of batteries tested is...
...one?
...two?
...three?
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Independent Events
Independent Events - in words: A and B are independent events if the
outcome of one does not affect the outcome of the other.
Examples of Independent Events:
Examples of Events Which are Not Independent:
Independent Events - in math: If A and B are independent events,
then
P (A | B) = P (A) and P (B | A) = P (B).
Test for the Independence of Two Events: Two events A and B are
independent if and only if
P (A ∩ B) = P (A) · P (B).
Example: If A and B are events with P (A) = .25, P (B) = .6 and P (A ∩
B) = .18.
• Are A and B independent?
• Are A and B mutually exclusive?
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Example & Test for Independence of More Than Two Events
There may be times where we want to test if more than two events are
independent. We can use the following rule.
Test for Independence of More than Two Events: If E1 , E2 , . . . , En
are independent events, then
P (E1 ∩ E2 ∩ . . . ∩ . . . En ) = P (E1 ) · P (E2 ) · · · · · P (En ).
Example: It is known that the events A, B, and C are independent. We
know that P (A) = 0.25, P (B) = 0.45 and P (C) = 0.5. Compute P (A ∩ B ∩ C).
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