Introductory Statistics

Lesson 4.1 B

Objective:

SSBAT construct a discrete probability distribution and its graph.SSBAT determine if a distribution is a probability distribution.

Standards: S2.5B

Review:

Discrete Random Variable

A random variable that has a finite or countable number of possible outcomes

The outcomes can be listed

The outcomes can be shown as just points on a number line

Examples: The number of students in a class

Discrete Probability Distribution

Lists each possible value the variable can be and its corresponding Probability

Must satisfy each of the following conditions

1. Each probability is between 0 and 1, inclusive

2. The sum of all the probabilities is 1

Example of a Discrete Probability Distribution

Days with Rain, x Probability P(x)

0 0.216

1 0.432

2 0.288

3 0.064

Determine if each is a Discrete Probability Distribution.Explain why or why not.

Each individual probability, P(x), has to be between 0 and 1, inclusive The sum of the probabilities has to equal 1

1. x P(x)5 0.286 0.217 0.438 0.15

No – The sum of all the probabilities is 1.07 not 1

Determine if each is a Discrete Probability Distribution.Explain why or why not.

Each individual probability, P(x), has to be between 0 and 1, inclusive The sum of the probabilities has to equal 1

2. x P(x)1 0.152 0.363 0.49

Yes – The sum of all the probabilities is 1 AND each individual probability is between 0 & 1

Determine if each is a Discrete Probability Distribution.Explain why or why not.

Each individual probability, P(x), has to be between 0 and 1, inclusive The sum of the probabilities has to equal 1

3. x P(x)1 0.52 0.253 0.754 -0.5

No – The probability of 4 is Negative

x 1 2 3 4 5P(x) 0.16 0.22 0.28 0.2  

Find the missing value in the probability distribution:

Remember that all the probabilities should add up to be 1

0.16 + 0.22 + 0.28 + 0.2 = 0.86

1 – 0.86 = 0.14

The missing value is 0.14

Constructing a DISCRETE Probability Distribution

1. Make a Frequency Distribution for the possible outcomes

2. Find the Sum of the frequencies

3. Find the probability of each possible outcome Divide the Frequency of each by the sum of

the frequencies

4. Check that each probability is between 0 and 1, inclusive, and that the sum is 1.

Example 1

An industrial psychologist administered a personality inventory test for passive-aggressive traits to 150 employees. Individuals were given a score from 1 to 5, where 1 was extremely passive and 5 was extremely aggressive. A score of 3 indicated neither trait. The frequency is shown below.

Score, x Frequency

1 24

2 33

3 42

4 30

5 21

Example 1 Continued

1st: Find the probability of each score Divide each individual score’s frequency by the total of the frequency column

Total of Frequencies = 150

P(1) = 24/150 = 0.16

P(2) = 33/150 = 0.22

P(3) = 42/150 = 0.28

P(4) = 30/150 = 0.20

P(5) = 21/150 = 0/14

Score, x Frequency

1 24

2 33

3 42

4 30

5 21

Example 1 Continued

Create the Probability Distribution using a table:

x

P(x)

x 1 2 3 4 5

P(x) 0.16 0.22 0.28 0.20 0.14

Example 2

A company tracks the number of sales new employees make each day during a 100-day probationary period. The results for one new employee are shown in the table below.

Construct a Discrete Probability Distribution.

Sales per dayx

Number of Days, f

0 16

1 19

2 15

3 21

4 9

5 10

6 8

7 2

Example 2 Continued

1st: Find the Probability of each x value

Total of Frequencies = 100

P(0) = 16/100 = 0.16

P(1) = 19/100 = 0.19

P(2) = 15/100 = 0.15

P(3) = 21/100 = 0.21

P(4) = 9 /100 = 0.09

P(5) = 10/100 = 0.10

P(6) = 8/100 = 0.08

P(7) = 2/100 = 0.02

Sales per dayx

Number of Days, f

0 16

1 19

2 15

3 21

4 9

5 10

6 8

7 2

Example 2 Continued

Create the Probability Distribution using a table:

x

P(x)

0 1 2 3 4 5 6 7

0.16 0.19 0.15 0.21 0.09 0.10 0.08 0.02

Complete Worksheet 4.1B