Probability Distributions

Problem: Suppose you are taking a true or false test with 6 questions… 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 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

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: 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 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