Chapter 7: Probability

Lesson 6: Probability Distributions

Mrs. Parziale

Vocabulary

• Random variable – is a variable whose values are numbers determined by the outcome of an experiment.

• Probability distribution – is a function which maps each value of a random variable onto its probability.

• Relative frequency – is a function which maps each value of a random variable onto its relative frequency (values found from actual data).

Example 1:

• Look at the probabilities when you toss two die and look at the sum of the numbers:

x = sum of dice

2 3 4 5 6 7 8 9 10 11 12

P(x) =

Graph this situation:

• What is the mean value?

• If you roll the dice again, what sum is most likely to occur?

6

4

2

5 10

Example 2:

• Suppose two die are rolled nine times. The sums of the rolls are: 6, 9, 10, 4, 8, 6, 11, 6, 8.

a. Multiply each random variable (x) by its probability P(x) and add them.

b. Find the mean of the numbers rolled.

x = sum of dice 4 6 8 9 10 11

P(x) =

Definition:

Let be a probability distribution.

The mean or expected value ( ) of the distribution is

_____________________________.

1 1 2 2{( , ( )), ( , ( ),....(( , ( ))}n nx P x x P x x P x

1

( ( ))n

i ii

x P x

x = sum of dice

2 3 4 5 6 7 8 9 10 11 12

P(x) =

Example 2, cont.

Consider the data (relative frequencies) from the experiment done in example 2.

a. Calculate the expected value of the probability distribution (using the original theoretical example).

1

36

4

36

5

36

6

36

5

36

4

36

3

36

2

36

1

36

2

36

3

36

Question -

Does the mean match the expected value of the theoretical probability distribution?

b. Find the percent error between mean of the relative frequencies and expected value of the theoretical distribution.

Probability Distribution Activity

1. Toss 2 dice 78 times.2. Record the sum of dice on the chart.3. Graph your data –

a. Correction – change the 36 denominators to 78 on the y-axis.

b. Graph the sums on the x-axis and the relative frequencies on the y-axis.

c. Answer the questions

4. Find the percent error.