1. Unit 4: Probability Distributions and Predictions 4.1 Probability Distributions and Expected Value

2. Posing and refining questions Collecting data and finding information Making sense of the data Making the case Evaluating the conclusions

3. A certain section of a course has a 4% failure rate. Is that good or bad? The results of a clinical trial of a certain test of a disease show a 15% false positive rate. If you are tested positive for the disease, should you followed the prescribed medical treatment?

4. Probability Distributions The last unit looked at probability of individual outcomes of an experiment Now, we will look at the distribution of probabilities of all possible outcomes Distributions can involve outcomes with equal or uneven probabilities

5. Experiment You will receive 2 dice Roll the two dice and record the sum n times We will combine the class results Calculate the experimental probability for each sum Create a probability distribution table of the possible sums

6. Sum of Two Dice Experimental Probability Distribution Theoretical Probability Distribution X23456789101112 P(X=x) X23456789101112 P(X=x)

7. Probability Distributions Provides probability of each possible value of random variable X Can be given in table or graph form Relative frequency distribution shows ratio of frequency to total number of trials

8. Variables Many probability experiments have numerical outcomes Random variable, X –Single value (denoted x) for each outcome Discrete variables –Values are counted Continuous variables –Infinite number of possible values on a continuous interval

9. Are the following variables discrete or continuous? Number of times you catch a ball Length of time you play ball Length of car in centimetres Number of red cars on highway Volume of water in a tank Number of candies in a box discrete continuous We will be looking at discrete random variables for now.

10. Uniform Probability Distribution All outcomes in distribution equally likely in any single trial Probability of discrete uniform distribution for all possible x

11. Expected Value, E(X) The value you would expect to get for one trial of an experiment –E.g.: The theoretical probability for sum of two dice is highest for 7 –You would expect to get a sum of 7 E(X) is the predicted average of all possible outcomes It’s actually just a weighted mean (the probabilities are the weights)!

12. E(X) of Sum of Two Dice

13. So? Why would we want to know the expected value of an experiment? We can compare the expected value to actual values Are the actual values “reasonable?” –That is, is that what we could expect to get?

14. Example 2 You pull 3 books out of your locker at random. You have 3 math/science books and 4 English/history books. a)What is the probability that at least two of the books are for a math/science class? b)Create a probability distribution table for the possible book selection. c)How many math/science books could you expect to pull out of your locker?

15. Example 2a sol’n Therefore there is a 37.1% chance of pulling out at least two math/science books.

16. Example 2b sol’n Combina- tions Proba- bility 0.343 0 math/sci 3 Eng/hist 1 math/sci 2 Eng/hist 2 math/sci 1 Eng/hist 3 math/sci 0 Eng/hist Calculations: 0.1140.0290.514 What is the sum of all the probabilities?1

17. Example 2c sol’n What is the discrete random variable X? X is the number of math/science books pulled out of the locker. You can expect to pull out about 1.3 math/science books.

18. What is the expected value of a fair game? 0 Why? The probabilities should be evenly distributed and the wins should balance the losses No game in a casino is fair. The expected value on a simple bet of a game of roulette is -$0.053. That is, for every spin, the casino expects to make about 5 cents. (How many spins are there in an average year?)

19. Roulette Odds Result of spinxP(X = x) Win (ball lands on your number 1-36) $35 Lose (ball lands on other number or 0 or 00) -$1