Transcript Binomial Probability Distributions

Probability Distributions
Problem: Suppose you are taking a true
or false test with 6 questions…. But
you didn’t study at all
Take out a coin and a piece of paper –
you will flip your coin to answer the
following problems.
 Heads is true, tails is false.

STATISTICS 257 Final Exam –
Oh no!
Get out your coin and guess the following:
1. If a gambling game is played with
expected value 0.40, then there is a 40%
chance of winning.
… I lost my notebook…

2. If A and B are independent events and
P(A)=0.37, then P(A|B)= 0.37.
…the textbook is too heavy

3. If A and B are events then P(A) + P(B)
cannot be greater than 1.
... I’m never going to really need this
stuff anyway, right?

4. If P(A and B) = 0.60, then P(A) cannot
be equal to 0.40.
…why are the questions so long?

5. If a business owner, who is only
interested in the bottom line, computes
the expected value for the profit made in
bidding on a project to be -3,000, then
this owner should not bid on this project.
…. Oops, I left my calculator in my
locker

6. Out of a population of 1000 people,
600 are female. Of the 600 females 200
are over 50 years old. If F is the event of
being female and A is the event of being
over 50 years old, then P(A|F) is the
probability that a randomly selected
person is a female who is over 50.
So how did you do?
#1 – False
 #2 – True
 #3 – False
 #4 – True
 #5 – True
 #6 – False

Tally up your responses – Did you pass?
The Distribution of scores on the test –
why is it more likely to get 3 right than to
get 6 right?
Try to determine the following probabilities
when guessing your answers on a true or
false test:
1.
0 right
2.
1right
Try to determine the following probabilities
when guessing your answers on a true or
false test:
1.
0 right
4.
4 right
2.
1right
5.
5 right
3.
2 right
6.
6 right
4.
3 right
7.
x right
Probability Distribution for Guessing on 6
True or False Questions
http://www.mathsisfun.com/data/quincunx.html
The Binomial Probability Distribution

A binomial probability is an experiment
where we count the number of successful
outcomes over n independent trials
Question: Is guessing the answer on 6 true
/ false questions a binomial probability?
Calculating a Binomial Probability

In general, we can calculate a binomial
probability of x successes on n independent
trials as:
Eg) What is the probability of guessing 4 out of 6
answers on a true or false quiz?
Try the following:
You are shooting 8 free throws and you have
a 75% of scoring on each. What is the
probability that you will:
1. Score on 0 shots?
2. Score
on 1 shot?
Try the following:
You are shooting 8 free throws and you have
a 75% of scoring on each. What is the
probability that you will:
1. Score on 0 shots?
2. Score
on 1 shot?
3. Score
on 2 shots?
4. Score
on at least 2 shots?
You are shooting 8 free throws and you
have a 75% of scoring on each. What is the
probability that you will:
5. Score at least 7 shots?
6. Score 6 or 7 shots?
7. Score all of your shots except the last
one?
You are shooting 8 free
throws and you have a 75%
chance of scoring on each.

How many shots do you expect to score?
The Expected Value of a Binomial

In general, the expected value of a
binomial probability is given as:
Try: What is the expected value of
1. Guessing on 100 True / false questions?
2.
Rolling a dice 600 times and counting 6s?
3.
Shooting 200 baskets with a 75% chance of making each
one
Try the following:
Suppose that 2% of all calculators bought from
Dollarama are defective.
You randomly collect 20 of them.
What is the probability that:
1. None of them are defective?
2.
2 or more are defective?
3.
In a batch of 1500, how many do you expect to
be defective?
Summary:
What is a probability distribution?
How do you calculate a binomial probability?
What are two conditions that you need in
order to use a binomial probability
calculation?
Why do you multiply a binomial probability by
nCx?
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p. 385 #1, 2, 3, 5, 6bc, 7ab, 8ab, 15, 17 Challenge: 10, 11