MDM4U1
5.1 Probability Distribution
Probability Distribution:
In this chapter we will develop models for distributions that show the probabilities of all possible
outcomes in an experiment.
For example, in an experiment where a die is rolled, the probability distribution would include
the probability of each possible outcome (1, 2, 3, 4, 5, and 6).
In this example “the number rolled with the die” is the random variable.
Throughout this unit we will be looking at 4 types of probability distributions:
1)
2)
3)
4)
Uniform Distribution
Binomial Distribution
Geometric Distribution
Hypergeometric Distribution
This unit describes probability distributions involving discrete random variables.
Recall:
Discrete Variables:
Continuous Variables:
Example 1 Classify the following random variables as discrete or continuous.
a) Length of time you stay in class
b) Number of courses you take in a semester
Uniform distribution is the distribution of probabilities with equally likely outcomes.
Uniform random variable with ‘n’ outcomes
𝑃(𝑥) =
1
𝑛
Example 2 Determine whether the following experiments have a uniform probability
a) Guessing a person’s age
b) Rolling a number on a die
Example 3 Uniform Probability Distribution
Determine the probability distribution for rolling a number on a dice.
x
1
2
3
4
5
6
P(x)
Expected Value is the average of the outcomes.
“Expected value” or “Expectation”
𝐸(𝑥) = 𝑥1 𝑃(𝑥1 ) + 𝑥2 𝑃(𝑥2 ) + ⋯ + 𝑥𝑛 𝑃(𝑥𝑛 )
𝑛
𝐸(𝑥) = ∑ 𝑥𝑖 𝑃(𝑥𝑖 )
𝑖=1
Example 4 Given the following probability distribution, determine the expected value.
x
2
4
6
P(x)
0.4
0.1
0.5
Example 5
a) Determine the probability distribution for the sum rolled with two dice.
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
b) Determine the expected sum of two dice.
x
1
2
3
4
5
6
7
8
9
10
11
12
P(x)