1. TIMES 3 Technological Integrations in Mathematical Environments and Studies Jacksonville State University 2010

2. Life isn’t fair. We can’t change that… but we can measure it.

3. Suppose that two playing cards are dealt to you at random from a well shuffled regular deck of 52 cards. If either card is a diamond, or both are, you win a dollar; otherwise, you lose a dollar. Is this game fair? How can you tell?

4. Definition: Random Variable A random variable is a function or rule that associates a real number with each element in a sample space.

5. Definition: Discrete Probability Distribution A list of all possible values of the random variable, x, and the corresponding probabilities, p(x) Recall the basics of probability: Every probability has a value between 0 and 1 inclusive. The probability of the sample space equals 1. P(event occurs) = 1 – P(event doesn’t occur)

6. Expected Value of a Discrete Random Variable Let X be a random variable with a known probability distribution. The mean, μ, or expected value, E(x), of X is given by

7. Back to the question…. Note that the conditions for winning (either or both are diamonds) are more complicated that for losing (neither is a diamond). So let’s think about the probability of losing. P(1 st card is not a diamond) = 39/52 P(2 nd card is not a diamond) = 38/51 P(neither is a diamond) = (39/52)(38/51) or.559 So, P(either or both are diamonds)= 1 -.559 =.441

8. E(x) = (-1)(.559) + (1)(.441) = -.118 So, over time, the player would average losing 11.8 cents for each game played. X1 P(x).559.441

9. Try it on your calculator. Enter the x values and probabilities in separate lists. Define the probability list as a frequency or weight. Ask for 1-variable summary stats.

10. Note that the mean of the discrete probability distribution is -.118 and the standard deviation of.993 rounds off to $1. Recall that the standard deviation describes how far away from the mean you’re likely to be for the majority of the time. So, for this game you’ll generally be down about 12 cents give or take a dollar.

11. If you’re interested, the formula for the standard deviation of a discrete probability distribution is given by:

12. Here’s another one… Flip a coin in the air three times. On each flip you must call the side you think will land facing up. If you are right at least two out of three times, you win a dollar; otherwise you lose a dollar.

13. This one is fair. Surprised? Let’s look at a sample space. $ Payoff#correctPossible outcomes P(x) 0www1/8 1rww wrw wwr 3/8 12wrr rwr rrw 3/8 13rrr1/8 r = guess right w = guess wrong

14. x 11 P(x)1/83/8 1/8 E(x) = 0 SD = 1 This is a fair game. Over time, you’ll break even give or take $1.

15. After the Mercedes Marathon, 10 finishers were selected and allowed to draw a key from a box. One key started a new Mercedes worth $37,500. One by one, the keys were tried until one unlocked the SUV. What is a fair price for a key at different stages of the contest? keys 10987654321 X pppppppppp $37500 1/101/91/81/71/61/51/41/31/21 0 9/108/97/86/75/64/53/42/31/20 E(x) 3750416746875357625075009375125001875037500 σ 1125011785124021312213975150001623817678187500

16. The expected value (mean) of the distribution gives the “fair price” for a key at each stage in the contest. Fair price increases as the number of keys decrease. Standard deviation increases as the number of keys decrease. This indicates greater risk in selling the key. Suppose there had been a random drawing for one winner of the SUV out of the 2600 finishers. What is a fair price for one chance? xp 375001/2600 02599/2600 E(x)$14.42 σ$735.00

17. Roulette Bet onPayoffOdds 135:137:1 1,217:119:1 1,2,311:135:3 1,2,3,4 8:117:2 1 – 6 5:116:3 1 – 12 2:113:6 1 - 18 1:110:9

18. Bet $1 on 1 The reward is 35:1 (given as amount won relative to amount bet) The risk is 37:1 (given as odds against winning) Calculate the expected net winnings E(x) = (-1)(37/38) + (35)(1/38) = -.0526 Expect to lose slightly over a nickel for each $1 bet on average. x35 P(x)37/381/38

19. InspireData Introduction and Examples 1.Dice rolling experiment (math) 2.Alligators (science) 3.World Religions (social studies) 4.eSurveys

20. One last problem… A dealer shuffles five black cards and five red cards and spreads them out on the table. You choose any two you want. If both cards are red, or if both are black, you win $1, otherwise, you lose $1.

21. P(1 st card is Red and 2 nd card is Red) = (5/10)(4/9) P(1 st card is Black and 2 nd card is Black = (5/10)(4/9) P(RR or BB) = (20/90) + (20/90) = 4/9 So, P(anything else)= 5/9 E(x) = -.11 SD = 1 Expect to lose 11 cents on average per game played give or take a dollar.

22. tetrahedron (4), cube (6), octahedron (8), dodecahedron (12), icosahedron (20) The Platonic Solids are the only possible shapes for fair dice. Questions?