1. Discrete Probability Distributions
CHAPTER 5
Discrete Probability Distributions

2. Chapter 5 Overview Introduction 5-1 Probability Distributions
5-2 Mean, Variance, Standard Deviation , and Expectation
5-3 The Binomial Distribution

3. 5-1Probability Distributions

4. A random variable is a variable whose values are determined by chance.
Discrete R.V
Continuous R.V
Variables that can assume all values in the interval between any two given values .
have a finite No. of possible value or an infinite No. of values can be counted .
CH.(6)
CH.(5)

5. For example: the probability experiment of tossing 3 coins ,
Sample space= {HHH,HHT,HTH,HTT,TTH,THT,THH,TTT}
Let: X be a random variable for a number of heads .
X: 0, 1, 2, 3

6. A discrete probability distribution consists of the values a random variable can assume and the corresponding probabilities of the values.
Graph
Table

7. Example 5-2:
Represent graphically the probability distribution for the sample space for tossing three coins.
Table

8. Graph

9. Example:
In a family with three children , find the probability distribution of the number of children who will be girls?

10. Example 5-3: X
Number of games played 4 8 5 7 6 9 16 40
Solution :

11. Table Graph Number of games X 4 5 6 7 Probability P(X) 0.200 0.175
0.225
0.400
Graph

12. Two requirements for Probability Distribution :
Example 5-4: X 5 10 15 20
P(X)
1/5 X 1 2 3 4
P(X)
1/4
1/8
1/16
9/16 X 2 4 6
P(X)
-1.0
1.5
0.3
0.2 X 2 3 7
P(X)
0.5
0.3
0.4

13. 5-2:Mean, Variance, Standard Deviation, and Expectation

14. 1- Mean :

15. Example 5-6:
In a family with two children , find the mean of the number of children who will be girls?

16. Example 5-6:
If three coins are tossed , find the mean of the number of heads that occur ?

17. Example 5-8:
The probability distribution shown represents the number of trips of five nights or more that American adults take per year. (That is, 6% do not take any trips lasting five nights or more, 70% take one trip lasting five nights or more per year, etc.) Find the mean. .

18. 2- Variance
3- Standard deviation Or

19. Example 5-9:
Compute the variance and standard deviation for the probability distribution in Example 5–5. .

20. Example 5-10:
A box contains 5 balls. Two are numbered 3, one is numbered 4 and two are numbered 5. The balls are mixed and one is selected at random . After a ball is selected , its number is recorded . Then it is replace. If the experiment is repeated many times , find the variance and standard deviation of the numbers on the balls.

21. 3- Expectation: Example (5-12):
One thousand tickets are sold at 1$ each for a color television valued at 350$ . What is the expected value of the gain if you purchase one ticket?

22. Solution : Approach 1: Win Lose Gain (X) 349$ -$1 Probability(X)
1/1000
999/1000
Approach 2:

23. Example 5-13:
One thousand tickets are sold at $1 each for four prizes of $100, $50,$25, $10. after each prize drawing , the winning ticket is then returned to the pool of tickets . What is the expected value if you purchased two tickets?
Solution :
Gain (X)
$98
$48
$23 $8
-$2
P(X)
2/1000
992/1000

24. The Binomial Distribution

25. A binomial experiment is a probability experiment that satisfies the following four requirements:
There must be a fixed number of trials.
Each trial can have only two outcomes or outcomes that can be reduced to two outcomes . These outcomes can be considered as either success or failure.
The outcomes of each trial must be independent of one another .
The probability of success must remain the same for each trial .

26. The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a Binomial Distribution
Notation for the Binomial Distribution
P(S) The probability of success
P(F) The probability of failure
P(S)= p and P(F)=1-p=q
n The number of trials
X The number of successes in n trials
X=0,1,2,3………..n

27. Binomial Probability formula
Example 5-15:
A coin is tossed 3 times . Find the probability of getting exactly two heads.
There must be a fixed number of trials(Three).
Each trial can have only two outcomes (heads, tails)
The outcomes of each trial must be independent of one another .
The probability of success must remain the same for each trial p(heads)=1/2.
n=3 ,X=2, p=1/2, q=1-p=1-1/2=1/2

28. Example 5-16:
A survey found that one out of five Americans say he or she has visited a doctor in any given month. If 10 people are selected at random , find the exactly 3 will have visited a doctor last month.
Solution :

29. Example 5-17:
A survey from Teenage Research Unlimited found that 30% of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random , find the probability that at least 3 them will have part-time jobs.
n=5,p=0.3,q=0.7,X=3,4,5
P(at least three teenagers have part time jobs)
= =0.162

30. 1-Mean: 2-Variance: 3-standard deviation:
Mean, Variance,Standard deviation for the Binomial Distribution
1-Mean:
2-Variance:
3-standard deviation:

31. Example 5-21:
A coin is tossed 4 times .find the mean ,variance and standard deviation of the number of heads that will be obtained .
Solution :
n=4, p=1/2,q=1/2,

32. Example 5-22:
A die is rolled 360 times .find the mean,variance,standard deviation of the number of 4s that will be rolled .
Solution :
n=360,p=1/6, q=5/6

33. Exercises: Q: How many times the die is rolled when the
mean =60 for the number of 3s.
360 10
1/6 20
Q: A student takes a 5 question multiple choice quiz with 4 choices for each question . If the student guesses at random on each question , what is the probability that the student gets exactly 3 questions correct?
0.022
0.264
0.088
0.313
Anc.A
Anc.C