1. 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

2. 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

3. 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

4. Example of a Discrete Probability Distribution Days with Rain, xProbability P(x) 00.216 10.432 20.288 30.064

5. 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. xP(x) 50.28 60.21 70.43 80.15  No – The sum of all the probabilities is 1.07 not 1

6. 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. xP(x) 10.15 20.36 30.49  Yes – The sum of all the probabilities is 1 AND each individual probability is between 0 & 1

7. 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. xP(x) 10.5 20.25 30.75 4-0.5  No – The probability of 4 is Negative

8. x12345 P(x)0.160.220.280.2 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

9. 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.

10. 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, xFrequency 124 233 342 430 521

11. Example 1 Continued 1 st : 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, xFrequency 124 233 342 430 521

12. Example 1 Continued Create the Probability Distribution using a table: x P(x) x 12345 0.160.220.280.200.14

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 day x Number of Days, f 016 119 215 321 49 510 68 72

15. Example 2 Continued 1 st : 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 day x Number of Days, f 016 119 215 321 49 510 68 72

16. Example 2 Continued Create the Probability Distribution using a table: x P(x) 01234567 0.160.190.150.210.090.100.080.02

17. Complete Worksheet 4.1B