ppt03. Constructing Probability Distribution.pptx

Statistics and Probability

1. Constructing Probability Distributions

2. Lesson Objectives At the end of this lesson, you should be able to: Illustrate a probability distribution for a discrete random variable and its properties; Compute probabilities corresponding to a given random variable; and Construct the probability mass function of a discrete random variable and its corresponding histogram.

3. Entry Card Find the Probability of the following events. Event (E) Probability P(E) 1. Getting an even number in a single roll of a die. 2. Getting a sum of 6 when two dice are rolled. 3. Getting an ace when a card is drawn from a deck 4. The probability that all children are boys if a couple has three children.

4. Find the Probability of the following events. Event (E) Probability P(E) 5. Getting an odd number and a tail when a die is rolled and a coin is tossed simultaneously. 6. Getting a sum of 11 when two dice are rolled. 7. Getting doubles when two dice are rolled.

5. Activity 1: Number of Tails Supposed three coins are tossed. Let Y be the random variable representing the number of tails that occur. Find the probability of each of the values of the random variable Y.

6. Solution: Step 1: Determine the sample space. Let H represent head and T represent tail. The sample space for this experiment is: S = (TTT, TTH, THT, HTT, HHT, HTH, THH, HHH) Step 2: Count the number of tails in each outcome in the sample space and assign this number to this outcome.

7. Legend: Y = Random variable representing the number of Tails H = Head T = Tail Possible Outcome Value of the Random Variable Y TTT TTH THT HTT HHT HTH THH HHH 3 2 2 2 1 1 1

8. Step 3: There are four possible values of the random variable Y representing the number of tails. There are 0, 1, 2, and 3. Assign probability values P(Y) to each value of the random variable.

9. Number of Tails Y Probability P(Y) 1 2 3 Number of Tails Y Probability P(Y) 1 2 3 There are 8 possible outcomes and no tail occur once. There are 8 possible outcomes and 1 tail occur three times. There are 8 possible outcomes and 2 tails occur three times. There are 8 possible outcomes and 3 tails occur once.

10. Table 1.1 The Probability Distribution or the Probability Mass Function of Discrete Random Variable Y Number of Tails Y 1 2 3 Probability P(Y) Number of Tails Y 1 2 3 Probability P(Y)

11. Discrete Probability Distribution A discrete probability distribution or a probability mass function consists of values a random variable can assume and the corresponding probabilities of the values.

12. Activity 2: Number of Blue Balls Two balls are drawn in succession without replacement from an urn containing 5 red balls and 6 blue balls. Let Z be the random variable representing the number of blue balls. Construct the probability distribution of the random variable Z.

13. Solution: Step 1 Determine the sample space. Let B represent the blue ball and R represent the red ball. The sample space for this experiment is: S = (RR, RB, BR, BB) Step 2 Count the number of blue balls in each outcome in the sample space and assign this number to this outcome.

14. Legend: Z = Random variable representing the number of Blue balls B = Blue ball R = Red ball Possible Outcome Value of the Random Variable Z RR RB BR BB 1 1 2

15. Step 3: There are three possible values of the random variable Z representing the number of blue balls. These are 0, 1, and 2. Assign probability values P(Z) to each value of random variable.

16. Number of Blue Balls Z Probability P(Z) 1 2 Number of Blue Balls Z Probability P(Z) 1 2 There are 4 possible outcomes and no blue ball occur once. There are 4 possible outcomes and 1 blue ball occur twice. There are 4 possible outcomes and 2 blue ball occur once.

17. Table 1.2 The Probability Distribution or the Probability Mass Function of Discrete Random Variable Z Number of Blue Balls Z 1 2 Probability P(Z) Number of Blue Balls Z 1 2 Probability P(Z)

18. Histogram Histogram is a bar graph. To construct histogram for a probability distribution, follow these steps. Plot the values of the random variable along the horizontal axis. Plot the probabilities along the vertical axis.

19. Activity 3: Defective Cell Phones Supposed three cell phones are tested at random. Let D represent the defective cell phone and let N represent the non-defective cell phone. If we let X be the random variable for the number of defective cell phone, construct the probability distribution of the random variable X.

20. Solution: Step 1: Determine the sample space. Let D represent the defective cell phone and N represent the non-defective cell phone. The sample space for this example is: S = (NNN, NND, NDN, DNN, NDD, DND, DDN, DDD) Step 2: Count the number of defective cell phone in each outcome in the sample space and assign this number to this outcome.

21. Legend: D= Defective cell phone N= Non-defective cell phone X= Random variable representing the number of defective cell phones Possible Outcome Value of the Random Variable X NNN 0 NND 1 NDN 1 DNN 1 NDD 2 DND 2 DDN 2 DDD 3

22. Step 3: There are four possible values of the random variable X representing the number of defective cell phones. These are 0, 1, 2, and 3. Assign probability values P(X), to each value of the random variable.

23. Number of Defective Cell phones X Probability P(X) 1 2 3 Number of Defective Cell phones X Probability P(X) 1 2 3 There are 8 possible outcomes and no defective cell phone occur once. There are 8 possible outcomes and 1 defective cell phone occur three times. There are 8 possible outcomes and 2 defective cell phone occur three times. There are 8 possible outcomes and 3 defective cell phone occur once.

24. Table 1.3 The Probability Distribution or the Probability Mass Function of Discrete Random Variable X Number of Defective Cell phone X 1 2 3 Probability P(X) Number of Defective Cell phone X 1 2 3 Probability P(X)

25. Properties of a Probability Distribution The probability of each value of the random variable must be between or equal to 0 and 1. In symbol, we write it as . The sum of the probabilities of all values of the random variable must be equal to 1. In symbol, we write it as  

26. Quiz: Determine whether the distribution represents a probability distribution . Write PD if it represents a probability distribution and NPT if NOT. 1. 2. X 1 5 8 7 9 P(X) X 1 5 8 7 9 P(X) X 2 4 6 8 P(X) X 2 4 6 8 P(X)

27. 3. 4. 5. X 1 2 3 5 P(X) X 1 2 3 5 P(X) X 4 8 12 15 17 P(X) X 4 8 12 15 17 P(X) X 1 3 5 7 P(X) 0.35 0.25 0.22 0.12

28. INDIVIDUAL ACTIVITY: 1. A shipment of five computers contains two that are slightly defective. If a retailer receives three of these computer at random, list the elements of the sample space S using the letters D and N for defective and non-defective computers, respectively. To each sample point assign a value x of the random variable X representing the number of computers purchased by the retailer which are slightly defective.

29. INDIVIDUAL ACTIVITY 2. Using the sample space for rolling two dice, construct a probability distribution for the random variable X representing the sum of the numbers that appear. Draw the graph of the probability distribution. 3. The probabilities that a customer buys 2,3,4,5, or 6 items in a convenience store are 0.32, 0.12, 0.23, 0.18, and 0.15, respectively. Construct the data and draw a histogram of the distribution.

30. Thank you for listening…