1. PROBABILITY DISTRIBUTIONS DISCRETE RANDOM VARIABLES OUTCOMES & EVENTS Mrs. Aldous & Mr. Thauvette IB DP SL Mathematics

2. You should be able to…  Construct a discrete probability distribution  Calculate expected value from a discrete probability distribution  Apply the concept of expectation to a variety of situations, including games of chance.

3. Challenge  How does rolling a die and doubling the score differ from rolling two dice and adding the scores?

4. Random Variables  A random variable is a rule that assigns exactly one value to each point in a sample space for an experiment.  A random variable can be classified as being either discrete or continuous depending on the numerical values it assumes.  A discrete random variable may assume either a finite number of values or an infinite sequence of values.  A continuous random variable may assume any numerical value in an interval or collection of intervals.

5. Random Variables QuestionRandom Variable x Type Family x = Number of dependents in Discrete size family reported on tax return Distance from x = Distance in miles from Continuous home to store home to the store site Own dog x = 1 if own no pet; Discrete or cat = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s)

6. Probability Distributions  The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable.

7. For example: Probabilities of flipping x heads from two coin tosses X – is the random variable for the event ‘number of heads’ x – is the number of heads for the calculations Number of heads (x) 0 1 2 P(X=x) 1 / 4 1 / 4 + 1 / 4 1 / 4 = 1 / 2 Probability of the event X being ‘x’

8. Who wants some money?  Here are the rules. You throw a die and if you throw n, I will give you $n thousand.  Would you play?  Would you pay to play? Why?  If so, what is the maximum you would pay to play?  How much would you expect to make in the long term if you kept playing this game with me?

9. Expectation  The mean of the random variable X is called the expected value of X …it is written E(X)  The expected value of X is:

10. Expectation: Simple Example The expected value of X is Number of heads (x) 0 1 2 P(X=x) 1 / 4 1 / 4 + 1 / 4 1 / 4 = 1 / 2 Probability of the event X being ‘x’ E(X) = 0 x 1 / 4 + 1 x 1 / 2 + 2 x 1 / 4 = 1 “You would expect 1 head out of every 2 throws”

11. Now you try  Sasha’s pocket contains one $1 coin, one 50 c coin, and three 20 c coins. He selects two coins at random to place in a charity collection box. The random variable X represents the amount, in cents, that he puts in the box.  (a) Show that P(X = 70) = 0.3.  (b) Find the probability distribution for X.  (c) Find the expected value of X.

12. Probability distribution tables 100 professional basketball players are surveyed. Each player has 5 3-point throws at a basket and the number of baskets scored is recorded in the table below. Baskets012345 Frequency10 253510 Using this table can you find the probability of a basketball player being picked at random scoring, a) 1 basket, b) 2 basket, c) 3 baskets, d) 0 baskets? Turn this into a probability distribution table...

13. Probability tables Baskets012345 Frequency10 253510 Baskets012345 Probability 0.1 0.250.350.1 From the table of probabilities we can find the average expected number of baskets: This is called the expected value, or E(x). You can also use a GDC, List 1: the baskets, List 2: the frequencies.

14. Another example The random variable X has a probability function given by the formula, a) Complete the table below to show the probabilities of x. b) Use your table to find the probability that x>4. c) Find E(x), the expected outcome of x. x23456 P(X=x) substitute 2 into the formula: now complete the table.

15. Question 1. The table below shows the probabilities of a random function, x. x123456 Pr(X=x)0.20.25p0.050.1 a) Find the value of p. b) Hence find E(x). p=0.3 E(x)=2.9

16. Probability Distributions Here are 2 of the probability distributions from the questions you have done so far. What do you notice? x123456 P(X=x)0.20.250.30.050.1 Number of heads (x) 0 1 2 P(X=x) 1 / 4 1 / 4 + 1 / 4 1 / 4 = 1 / 2 Probability of the event X being ‘x’

17. Make your own probability distribution Place all the numbers from the bag into the table below (with no repeats!) to make a viable probability distribution. How many can you make?

18. Question A die is biased such that the probabilities of the die landing on each score is given below: (a) Show that 5x + 10y = 0.9 (b) Given that E(S) = 3.35, find the values of x and y.

19. Exam Style Question Two fair four-sided dice, one red and one green, are thrown. For each die, the faces are labelled 1, 2, 3, 4. The score for each die is the number that lands face down. (a) Write down (i) the sample space (ii) the probability that two scores of 4 are obtained

20. Exam Style Question Continued… (a) Write down (i) the sample space

21. (a) Write down (ii) the probability that two scores of 4 are obtained

22. Exam Style Question Continued… Let X be the number of 4s that land face down. (b) Copy and complete the following probability distribution for X.

23. Let X be the number of 4s that land face down.

24. Exam Style Question Continued… (c) Find E(X).

25. You should know…  A random variable, denoted by X, takes on observed values x 1, x 2, x 3, …, x n, and if the data obtained can be counted, then X is a discrete random variable  The probability that the random variable X can take on the values x i is denoted by P(X = x i )  A discrete probability distribution lists the values of the random variable with their corresponding probabilities

26. You should know…  A probability distribution may be presented in tabular form or as a function. For example,  The sum of all probabilities in a distribution is 1

27. You should know…  The expected value of a distribution is its mean and is denoted by E(X) or. It is determined by summing the product of the outcomes with their corresponding probabilities, that is,

28. Be prepared…  Although expected value is an average, there is no need to divide by n as the formula for expected value already takes this into account.