Probability Distributions

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# Probability Distributions - PowerPoint PPT Presentation

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!.

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

0 right

1right

2 right

3 right

4 right

5 right

6 right

x right

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:

• Score on 0 shots?
• 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:

• Score on 0 shots?
• Score on 1 shot?
• Score on 2 shots?
• 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

• Guessing on 100 True / false questions?
• Rolling a dice 600 times and counting 6s?
• 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:

• None of them are defective?
• 2 or more are defective?
• 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?

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