1. Introduction to Probability and Statistics
Chapter 5
Discrete Distributions

2. Discrete Random Variables
Discrete random variables take on only a finite or countable many of values.
Number of heads in 1000 trials of coin tossing
Number of cars that enter UNI in a certain day

3. Binomial Random Variable
The coin-tossing experiment is a simple example of a binomial random variable. Toss a fair coin n = 3 times and record
x = number of heads. x
p(x)
1/8 1
3/8 2 3

4. Example Toss a coin 10 times
For each single trial, probability of getting a head is 0.4
Let x denote the number of heads

5. The Binomial Experiment
The experiment consists of n identical trials.
Each trial results in one of two outcomes, success (S) or failure (F).
Probability of success on a single trial is p and remains constant from trial to trial. The probability of failure is q = 1 – p.
Trials are independent.
Random variable x, the number of successes in n trials.
x – Binomial random variable with parameters n and p

6. Binomial or Not? A box contains 4 green M&Ms and 5 red ones
Take out 3 with replacement
x denotes number of greens
Is x binomial?
Yes,
3 trials are independent with same probability of getting a green. 6

7. Binomial or Not? A box contains 4 green M&Ms and 5 red ones
Take out 3 without replacement
x denotes number of greens
Is x binomial?
NO, when we take out the second M&M, the probability of getting a green depends on color of the first.
3 trials are dependent.

8. Binomial or Not?
Very few real life applications satisfy these requirements exactly.
Select 10 people from the U.S. population, and suppose that 15% of the population has the Alzheimer’s gene.
For the first person, p = P(gene) = .15
For the second person, p  P(gene) = .15, even though one person has been removed from the population…
For the tenth person, p  P(gene) = .15
Yes, independent trials with the same probability of success

9. Binomial Random Variable
Rule of Thumb:
Sample size n; Population size N;
If n/N < .05, the experiment is Binomial.
Example: A geneticist samples 10 people and x counts the number who have a gene linked to Alzheimer’s disease.
Success:
Failure:
Number of
trials:
Probability of Success
n = 10
Has gene
p = P(has gene) = 0.15
Doesn’t have gene

10. Example Toss a coin 10 times
For each single trial, probability of getting a head is 0.4
Let x denote the number of heads
Find probability of getting exactly 3 heads. i.e. P(x=3).
Find probability distribution of x

11. Solution Simple events: Event A: {strings with exactly 3 H’s};
Strings of H’s and T’s with length 10
Simple events:
Event A: {strings with exactly 3 H’s};
HTTTHTHTTT TTHHTTTTHT…
Probability of getting a given string in A:
HTTTHTHTTT
Number of strings in A
Probability of event A. i.e. P(x=3)

12. A General Example
Toss a coin n times; For each single trial, probability of getting a head is p;
Let x denote the number of heads;
Find the probability of getting exactly k heads. i.e. P(x=k)
Find probability distribution of x.

13. Binomial Probability Distribution
For a binomial experiment with n trials and probability p of success on a given trial, the probability of k successes in n trials is

14. Binomial Mean, Variance and Standard Deviation
For a binomial experiment with n trials and probability p of success on a given trial, the measures of center and spread are:

15. Example A marksman hits a target 80% of the
time. He fires 5 shots at the target.
What is the probability that exactly 3 shots hit the target?
n =
p =
x =
success =
hit .8
# of hits 5

16. Example
What is the probability that more than 3 shots hit the target?

17. Example x = number of hits.
What are the mean and standard deviation for x? (n=5,p=.8) m

18. Cumulative Probability
You can use the cumulative probability tables to find probabilities for selected binomial distributions.
Binomial cumulative probability:
P(x  k) = P(x = 0) +…+ P(x = k) 18

19. Key Concepts I. The Binomial Random Variable 1. Five characteristics:
the experiment consists of n identical trials;
each resulting in either success S or failure F;
probability of success is p and remains constant;
all trials are independent;
x is the number of successes in n trials.
2. Calculating binomial probabilities
a. Formula:
b. Cumulative binomial probability P(x  k).
3. Mean of the binomial random variable:
4. Variance and standard deviation:

20. Example According to the Humane Society of the
United States, there are approximately 40% of
U.S. households own dogs. Suppose 15
households are selected at random. Find
probability that exactly 8 households own dogs?
probability that at most 3 households own dogs?
probability that more than 10 own dogs?
the mean, variance and standard deviation of x, the number of households that own dogs.

21. Example
According to the Humane Society of the United States, there are approximately 40% of U.S. households own dogs. Suppose 15 households are selected at random.
What is probability that exactly 8 households own dogs?
n =
p =
x =
success = 15
own dog .4
# households that own dog

22. Example
What is the probability that at most 3 households own dogs?

23. Example
What are the mean, variance and standard deviation of random variable x?
(n=15, p=.4)

24. Binomial Probability
Probability distribution for Binomial random variable x with n=15, p=0.4

25. Example
What are the mean, variance and standard deviation of random variable x?
Calculate interval within 2 standard deviations of mean. What values fall into this interval?
Find the probability that x fall into this interval.