Unit 7
Probability Distributions
Probability Distributions
Probability Distributions
In the last unit, the emphasis was on the probability
of individual outcomes. In this unit we will be
looking at the distributions of all possible outcomes
of a probability experiment
• Many probability experiments have numerical
outcomes that can be counted or measured
• A random variable, X, has a single value for
each outcome in an experiment
• Discrete random variables have values
separate from one another (eg. The number of
phone calls made by a salesperson)
• Continuous random variables have an infinite
number of possible values in a continuous
interval (eg. The length of time a salesperson
spent on the phone)
Probability Distribution – a table, formula, or
graph that provides the probabilities of all
possible values of a discrete random variable
assuming any of its possible values
Example 1
Determine the probability distribution for rolling a
standard 6-sided die.
Outcomes of rolling a die are discrete random variables.
Let X rep the value on the face of the die.
x
1
2
3
4
5
6
Sum
P(x)
• Since the probability for
each outcome is equal,
this is a uniform
probability distribution.
• The sum of the
probabilities in any
probability distribution is 1.
The probabilities can also be represented as a
graph.
Probabilities for Rolling a Die
0.2
P(x)
0.15
0.1
0.05
0
1
2
3
4
x
5
6
Example 2
Determine the probability distribution for the
number of tails in three tosses of a coin.
Let X be the number of tails tossed.
X
0
1
2
3
Probability Distribution
P(X)
If you repeated this experiment (tossing three coins)
many times, on average how many tails would you
expect to get?
• After many repetitions of an experiment, the
average (mean) value of the random variable
tends towards the expected value, E(X)
• The expected value is calculated using a
weighted mean formula:
𝐸 𝑋 =
𝑥𝑃(𝑥)
Example 3
Suppose you toss three coins. What is the
expected number of tails?
Let X rep the number of tails.
We already calculated the probability distribution
for this experiment.
𝐸 𝑋 =
𝑥𝑃(𝑥)
1
3
3
3
=0
+1
+2
+3
8
8
8
8
= 1.5
The expected number of tails is 1.5
X
0
1
2
3
P(X)
1
8
3
8
3
8
1
8
Example 4
Suppose you want to select a committee consisting of
three people. The group from which the committee
members can be selected consists of four men and
three women.
a. What is the probability that there will be 1
woman on the committee?
2 of 4 males selected
1 of 3 females selected
4𝐶2
× 3𝐶1
=
7𝐶 3
Total number of committees:
3 of 7 people selected
b. Determine the probability distribution for the
number of women on the committee.
Let X represent the number of women on the
committee.
X
0
1
2
3
P(X)
c. What is the expected number of women on the
committee?
𝐸 𝑋 =
=0
𝑥𝑃(𝑥)
4
35
+1
18
35
+2
12
35
+3
1
35
= 1.3
The expected number of women is 1.3.
Example 5
Consider a dice game in which you roll a single die. If
you roll an even number you gain that number of
points. If you roll an odd number you lose that number
of points.
a. Determine the probability distribution for this game.
Let X represent
the number rolled.
x
1
2
3
4
5
6
Points
P(x)
b. What is the expected number of points for this
game?
𝐸 𝑋 =
𝑥𝑃(𝑥)
1
1
1
1
1
1
= −1
+2
−3
+4
−5
+6
6
6
6
6
6
6
= 0.5
The expected score is 0.5
c. Is this a fair game?
• This game is not fair because the points gained
and lost are not equal.
• In order for a game to be fair, the expected value
must equal 0.
Practice
• Page 376
# 1, 2, 3, 4, 6ab, 8, 9, 10, 11, 17