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Bluman, Chapter 5 1
Test 4 and 5Wednesday Oct 30th
guessing
Suppose there is multiple choice quiz on a subject you don’t know anything about…. 15th Century Russian Literature; Nuclear physics etc.
You have to guess on every question. There are 5 questions and each question
has 4 choices.
Bluman, Chapter 5 2
Let x be the score on the test.Find p(x=0)
In another words the probability you will get a score of zero, i.e. you will get all the questions wrong Find p(x=1)
In another words the probability you will get a score of 1, i.e. you will get only one question correct.
Bluman, Chapter 5 3
Bluman, Chapter 5 4
Question number
Correct or wrong
1
2
3
4
5
Repeat the process:
P(2)=
P(3)=
p(4)=
P(5)=
Bluman, Chapter 5 5
What if the number of questions changed Let’s say now the test has 10 questions
and each question has 4 choices. What does the probability distribution chart
looks like?
Bluman, Chapter 5 6
Bluman, Chapter 5 7
x P(x)
012345678910
What if the number of choices changes Let’s say now the test has 10 questions
and each question has 5 choices. What does the probability distribution chart
looks like?
Bluman, Chapter 5 8
Bluman, Chapter 5 9
12345678910
5-3 The Binomial Distribution
10
Many types of probability problems have only two possible outcomes or they can be reduced to two outcomes.
Examples include: when a coin is tossed it can land on heads or
tails,
when a baby is born it is either a boy or girl.
It will rain or it won’t
A person will pass the bar exam or not.
The Binomial Distribution
Bluman, Chapter 5 11
The binomial experiment is a probability experiment that satisfies these requirements:
1. Each trial can have only two possible outcomes—success or failure.
2. There must be a fixed number of trials.
3. The outcomes of each trial must be independent of each other.
4. The probability of success must remain the same for each trial.
Notation for the Binomial Distribution
Bluman, Chapter 5 12
The symbol for the probability of success
The symbol for the probability of failure
The numerical probability of success
The numerical probability of failure
and P(F) = 1 – p = q
The number of trials
The number of successes
P(S)
P(F)
p
q
P(S) = p
n
X
Note that X = 0, 1, 2, 3,...,n
The Binomial Distribution
!
- ! ! X n Xn
P X p qn X X
Bluman, Chapter 5 13
In a binomial experiment, the probability of exactly X successes in n trials is
number of possible probability of adesired outcomes desired outcome
or
X n Xn xP X C p q
Chapter 5Discrete Probability Distributions
Section 5-3Example 5-16
Page #272
Bluman, Chapter 5 14
Example 5-16: Survey on Doctor Visits
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 probability that exactly 3 will have visited a doctor last month.
Bluman, Chapter 5 15
!
- ! ! X n Xn
P X p qn X X
3 7
10! 1 43
7!3! 5 5
P
1510,"one out of five" , 3 n p X
0.201
Chapter 5Discrete Probability Distributions
Section 5-3Example 5-17
Page #273
Bluman, Chapter 5 16
Example 5-17: Survey on EmploymentA survey from Teenage Research Unlimited (Northbrook, Illinois) 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 of them will have part-time jobs.
Bluman, Chapter 5 17
3 25!3 0.30 0.70
2!3! P
5, 0.30,"at least 3" 3,4,5 n p X
0.132
4 15!4 0.30 0.70
1!4! P 0.028
5 05!5 0.30 0.70
0!5! P 0.002
3 0.132
0.028
0.002
0.162
P X
Chapter 5Discrete Probability Distributions
Section 5-3Example 5-18
Page #273
Bluman, Chapter 5 18
Example 5-18: Tossing CoinsA coin is tossed 3 times. Find the probability of getting exactly two heads, using Table B.
Bluman, Chapter 5 19
123, 0.5, 2 n p X 2 0.375 P
The Binomial Distribution
Mean: np2Variance: npq
Bluman, Chapter 5 20
The mean, variance, and standard deviation of a variable that has the binomial distribution can be found by using the following formulas.
Standard Deviation: npq
Chapter 5Discrete Probability Distributions
Section 5-3Example 5-23
Page #276
Bluman, Chapter 5 21
Example 5-23: Likelihood of TwinsThe Statistical Bulletin published by Metropolitan Life Insurance Co. reported that 2% of all American births result in twins. If a random sample of 8000 births is taken, find the mean, variance, and standard deviation of the number of births that would result in twins.
Bluman, Chapter 5 22
8000 0.02 160 np
2 8000 0.02 0.98 156.8 157 npq
8000 0.02 0.98 12.5 13 npq
Tech notes
Read technology notes on page 281.
Read example 5-19 on page 274
Exercises 5.3
Page 276 #1, 5, 11, 15 and 17
Bluman, Chapter 5 23