11.3 – Conditional Probability – Events Involving “And”
Conditional Probability
The probability of an event based on the fact that some other event has
occurred, will occur, or is occurring.
The probability of event B occurring given that event A has occurred is
usually stated as “the conditional probability of B, given A; P(B/A)
P( A and B)
P( A  B)

P(B/A) =
P( A)
P( A)
11.3 – Conditional Probability – Events Involving “And”
Conditional Probability
Example:
A number from the sample space S = {2, 3, 4, 5, 6, 7, 8, 9} is randomly
selected. Given the defined events A and B,
A: selected number is odd, and
B: selected number is a multiple of 3
find the following probabilities.
a) P(B)
b) P(A and B)
c) P(B/A)
a) B = {3, 6, 9}
P(B) = 3/8
b) P(A and B) = P({3, 5, 7, 9}  {3, 6, 9})
= P({3, 9}) = 2/8 = 1/4
c) Probability of B given A has occurred:
P(B/A) =
P(A and B)
P(A)
=
1/4
4/8
= 1/2
11.3 – Conditional Probability – Events Involving “And”
Conditional Probability
Example:
Given a family with two children, find the probability that both are boys,
given that at least one is a boy.
Conditional Probability
S= {gg, gb, bg, bb}
A = at least one boy
B = both are boys
P( A and B)
P(B/A) =
P( A)
A = {gb, bg, bb}
B = {bb}
P(A and B) = P({gb, bg, bb}  {bb}) = P({bb}) = 1/4
P(A) = P({gb, bg, bb}) = 3/4
P( A and B)
=
P( A)
1/4
3/4
= 1/3
11.3 – Conditional Probability – Events Involving “And”
Independent Events
Two events are Independent if the occurrence of one of them has no
effect on the probability of the other.
P(B/A) = P(B)
or
P(A/B) = P(A)
11.3 – Conditional Probability – Events Involving “And”
Independent Events
Example:
A single card is randomly selected from a standard 52-card deck. Given
the defined events A and B,
A: the selected card is an ace,
B: the selected card is red,
find the following probabilities.
a) P(B)
a) P(B) =
26
52
b) P(A and B)
c) P(B/A)
= 1/2
b) P(A and B) = P({Ah, Ad, Ac, As}  {all red}) = P({Ah, Ad}) = 2/52
c) P(B/A) =
P(A and B)
P(A)
=
2/52
4/52
= 1/2
Events A and B are independent as P(B) = P(B/A).
11.3 – Conditional Probability – Events Involving “And”
Multiplication Rule of Probability - Events Involving “And”
If A and B are any two events then
P(A and B) = P(A)  P(B/A)
If A and B are independent events then
P(A and B) = P(A)  P(B)
Example:
A jar contains 4 red marbles, 3 blue marbles, and 2 yellow marbles.
What is the probability that a red marble is selected and then a blue one
without replacement?
P(Red and Blue) = P(Red)  P(Blue/Red)
= 4/9  3/8
= 12/72
= 1/8
= 0.1667
11.3 – Conditional Probability – Events Involving “And”
Multiplication Rule of Probability - Events Involving “And”
Example:
A jar contains 4 red marbles, 3 blue marbles, and 2 yellow marbles.
What is the probability that a red marble is selected and then a blue one
with replacement?
P(Red and Blue) = P(Red)  P(Blue)
= 4/9  3/9
= 12/81
= 4/27 = 0.148
11.5 – Expected Value
The Expected Value of x is the sum of the products of the
values of x and their corresponding probabilities.
E(x) = x1 P(x1) + x2 P(x2) + x3 P(x3) + … + xn P(xn)
The expected value is a calculation that serves as the best
prediction of a value. It is the probability-weighted average of
all possible outcomes.
The expected value of a possible future event assists in making
mathematically sound decisions. It is often used when making
investments, determining a price for numerous services,
prioritizing events, and in calculating Return on Investment.
11.5 – Expected Value
Example:
A third grade class was surveyed regarding the number of hours that they
played electronic games each day. The probability distribution is given
in the table below:
# of Hours (x)
Probability P(x)
0
0.3
1
0.4
2
0.2
3
0.1
Calculate the Expected Value of the quantity of time that a third grader
spends each day playing electronic games.
11.5 – Expected Value
E(x) = x1 P(x1) + x2 P(x2) + x3 P(x3) + … + xn P(xn)
# of Hours (x)
Probability P(x)
0
0.3
1
0.4
2
0.2
3
0.1
Expected value, E(x) = 0 (0.3) + 1 (0.4) + 2 (0.2) + 3 (0.1)
Expected value, E(x) = 0 + 0.4 + 0.4 + 0.3
Expected value, E(x) = 1.1 hours
11.5 – Expected Value
Example:
Find the expected number of boys for a three-child family. Assume girls
and boys are equally likely.
bbb
# of Boys
Probability
Product
bbg
x
P(x)
x P(x)
bgb
0
1/8
0
bgg
1
3/8
3/8
gbb
2
3/8
6/8
gbg
3
1/8
3/8
ggb
ggg
Expected value, E(x) = 0 + 3/8 + 6/8 + 3/8
Expected value, E(x) = 12/8
Expected value, E(x) = 3/2
Expected value, E(x) = 1.5 boys
11.5 – Expected Value
Example:
Finding Expected Winnings
A player pays $3 to play the following game:
Win $7 by rolling a 6 on a single die,
Win $1 by rolling any other number.
What are the expected net winnings for the game?
Number
1, 2, 3, 4, 5
6
Payoff
Net
P(x)
x P(x)
11.5 – Expected Value
Example:
Finding Expected Winnings
A player pays $3 to play the following game:
Win $7 by rolling a 6 on a single die,
Win $1 by rolling any other number.
What are the expected net winnings for the game?
Number
Payoff
Net
P(x)
x P(x)
1, 2, 3, 4, 5
$1
- $2
5/6
- $10/6
6
$7
$4
1/6
$4/6
Expected value, E(x) = – $10/6 + $4/6
Expected value, E(x) = – $6/6
Expected value, E(x) = – $1
11.5 – Expected Value
A fair game is one in which the net winnings are zero.
An unfair game against the player has negative expected winnings.
An unfair game in favor of the player has positive expected winnings.
Previous Example:
Finding Expected Winnings
A player pays $3 to play the following game:
Win $7 by rolling a 6 on a single die,
Win $1 by rolling any other number.
Is this a fair game?
NO
It is an unfair game against the player as it has negative expected
winnings (–$1) .
11.5 – Expected Value
Example:
Expected Investment Profits
Mark intends to invest $6,000 in one of two companies. His research is
presented in the tables below:
Company ABC
Company PDQ
Profit/Loss (x)
Probability P(x)
Profit/Loss (x)
Probability P(x)
- $400
0.2
$600
0.8
$800
0.5
$1000
0.2
$1300
0.3
What are the expected profits (or loses) for each company?
E(x) = -400 (0.2) + 800 (0.5) + 1300 (.3)
E(x) = 600 (0.8) + 1000 (0.2)
E(x) = -80 + 400 + 390
E(x) = 480 + 200
E(x) = $710 Profit
E(x) = $680 Profit