Probability DistributionsProbability Distributions

Problem: Suppose you are Problem: Suppose you are taking a true or false test with taking a true or false test with 6 questions…. But you didn’t 6 questions…. But you didn’t study at allstudy at allTake 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 – STATISTICS 257 Final Exam – Oh no!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…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 the textbook is too heavyheavy3. If A and B are events then P(A)

+ P(B) cannot be greater than 1.

... I’m never going to really need ... I’m never going to really need this stuff anyway, right?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 why are the questions so long?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 . Oops, I left my calculator in my lockermy locker6. 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?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 The Distribution of scores on the test – why is it more likely to get 3 test – why is it more likely to get 3 right than to get 6 right?right than to get 6 right?

Try to determine the following Try to determine the following probabilities when guessing your probabilities when guessing your answers on a true or false test:answers on a true or false test:1. 0 right

2. 1right

Try to determine the following Try to determine the following probabilities when guessing your probabilities when guessing your answers on a true or false test:answers on a true or false test:1. 0 right

2. 1right

3. 2 right

4. 3 right

4. 4 right

5. 5 right

6. 6 right

7. x right

Probability Distribution for Guessing on 6 True or False Questions

http://www.mathsisfun.com/data/quincunx.html

The Binomial Probability The Binomial Probability DistributionDistributionA 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 Calculating a Binomial ProbabilityProbabilityIn 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: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: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 are shooting 8 free throws and you have a 75% of scoring on each. you have a 75% of scoring on each. What is the probability that you What is the probability that you will: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 You are shooting 8 free throws and you have a throws and you have a 75% chance of scoring 75% chance of scoring on each.on each.

How many shots do you expect to score?

The Expected Value of a The Expected Value of a BinomialBinomialIn general, the expected value of

a binomial probability is given as:

Try: What is the expected value of1. 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: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: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?

p. 385 #1, 2, 3, 5, 6bc, 7ab, 8ab, 15, 17 Challenge: 10, 11