Probability Distributions
(and Mathematical Expectation)
This unit develops models for distributions that
show the probabilities of all possible outcomes
of an experiment.
DEFINITIONS:
random variable (X):  the variable to be measured
 it has a single value for each outcome
in an experiment
 ex. If X is the number rolled with one
die, then x = 1, 2, 3, 4, 5, or 6
discrete random
variable:
continuous random
variable:
probability
distribution:
uniform probability
distribution:
 values that are separate from each
other (ie) integer values
 ex. the number of phone calls made
by a salesman
 an infinite number of values on a
continuous interval
 ex. the length of time spent on the
phone by a salesman
 shows the probabilities of all the
possible outcomes of an experiment
 the sum of the probabilities in any
distribution is 1
 all outcomes are equally likely
Example 1 A single die is rolled.
a) Define the random variable for the experiment.
b) Determine the probability distribution for the experiment.
Random
Variable (X)
Probability
P(x)
c) Draw the probability distribution graph.
P(x)
Probability
X
Number Rolled
Is this an example of a uniform probability distribution?
Example 2 Create a probability distribution table and a
probability distribution graph for the number of heads
that come up in the toss of two coins.
The random variable for the experiment is ________________.
S= {
}
Random
Variable (X)
Probability
P(x)
Is this an example of a uniform probability distribution?
EXPECTED VALUE (or Mathematical Expectation)
expected value E(X): the predicted average of all possible
outcomes of a probability experiment
E(X) = x1P(x1) + x2P(x2) + x3P(x3) + … + xnP(xn)
Example 1 Given the following probability distribution, determine
the expected value.
x
2
4
6
P(x)
0.42
0.37
0.21
Example 2 In a school lottery, 1500 tickets are sold.
PRIZES ($)
50
25
10
# of PRIZES
1
2
5
a) Determine the probability of winning a prize.
b) Determine the expected value.
c) If a ticket cost 50¢, determine the profit that the school
makes on each ticket.
Homework:
p.374 – 376
#1, 2, 3ac, 4, 6ab, 8 – 12, 16bcd
Additional Example
(Probability Distribution)
p. 375
#6.
Determine the probability distribution for the sum rolled
with two dice.
The random variable (x) is the sum of the two dice.
Sum
(x)
# of ways to achieve sum
P(x)
Example – A Two Dice Game
A player pays $7 and then rolls two dice.
The player is paid according to the given table.
a)
b)
What amount can a player expect to win/lose by playing this game?
What would be a “fair” value to pay to play this game?
Sum
2
3
4
5
6
7
8
9
10
11
12
Winnings
$ 10
9
8
7
6
5
6
7
8
9
10
A game is fair when E(x)=0.
This is also the “break–even” point!