Randomness, Probability, and Simulation

1. Randomness, Probability, and Simulation Honors Statistics

2. Randomness, Probability, and Simulation Learning Objectives Assessment Opportunity After this section, you should be able to… • DESCRIBE the idea of probability • DESCRIBE myths about randomness • CHECK YOURSELF Problems • TICKET OUT

3. Genetics Problem for Ticket Out Suppose a married man and woman both carry a gene for cystic fibrosis but don’t have the disease themselves. According to the laws of genetics, the probability that their first child will develop cystic fibrosis is 0.25. • Explain what this probability means in context of the situation. • Why doesn’t this probability say that if the couple has 4 children, one of them is guaranteed to get cystic fibrosis?

4. Chance • We are surrounded by chance • A coin toss at the football game decides who will receive the ball first • Lottery • Casinos • Genetic traits that children inherit such as gender, hair and eye color, blood type, etc. are determined by chance

5. Chance and Probability • Although many outcomes are governed by chance, patterns emerge after many repetitions • We use mathematics to understand the regular pattern of chance behavior when we repeat the same chance process again and again • The mathematics of chance is called probability

6. 1 in 6 wins As a special promotion for its 20 ounce bottles of soda, a soft drink company printed a message on the inside of each bottle cap. Some of the caps said, “Please try again!” wile others said, “You’re a winner!” The company advertised the promotion with the slogan “1 in 6 wins a prize.” Seven friends each by one 20 ounce bottle at a local convenience store. The store clerk is surprised when three of them win a prize. Is this group of friends just lucky, or is the company’s 1-in-6 claim inaccurate?

7. 1 in 6 Simulation (Work in Pairs) 1. Assume the company is telling the truth 2. Let 1 through 5 represent “Please try again!” and 6 represent “You’re a winner.” 3. Roll your dice seven times to imitate the process of the 7 friends buying their sodas. How many of them won a prize? 4. Repeat step 3 four more times. In your five repetitions of this simulation, how many times did three or more of the group win a prize? 5. Plot your results on the dot plot on the board 6. Based on the results, does it seem plausible that the company is telling the truth, but that the seven friends just got lucky?

8. PLOT NUMBER OF WINNERS ____________________________________________________ 0 1 2 3 4 5 6 7 Number of Winners

9. Coin Toss Applet • http://bcs.whfreeman.com/tps4e/#628644__666397__

10. Randomness, Probability, and Simulation • The Idea of Probability Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. The law of large numbers says that if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value. Definition: The probability of any outcome of a chance process is a number between 0 (never occurs) and 1(always occurs) that describes the proportion of times the outcome would occur in a very long series of repetitions.

11. Probability • Probability answers the question “What would happen if we did this many times?”

12. Life InsuranceProbability and Risk How do life insurance company’s decide how much to charge for life insurance? According to the National Center for Health Statistics • Proportion of men aged 20 to 24 who will die in any one year is 0.0015. This is the probability of death. • The probability of death for a woman 20 to 24 is 0.0005 • Company charges the men 3 times as much as the women because the company knows it will have to pay on about .15% of the policies for men and .05% of the policies sold to women

13. Extended Warranties How much should a company charge for an extended warranty for a specific type of cell phone? • 5% of the phones under warranty will be returned • Cost to replace the phone is $150 They should charge at least (.05)(150)= $7.50

14. Check Your Understanding 1, According to the “Book of Odds,” the probability that a randomly selected U.S. adult usually eats breakfast is 0.61. • Explain what probability 0.61 means in this setting • Why doesn’t this probability say that if 100 U.S. adults are chosen at random, exactly 61 of them will eat breakfast? 2. Probability is a measure of how likely an outcome is to occur. Match one of the probabilities that follow with each statement. 0 0.01 0.3 0.6 0.99 1 • This outcome is impossible. It can never occur. • This outcome is certain. It will occur on every trial • This outcome is very unlikely, but will occur once in a while in a long sequence of trials • This outcome will occur more often than not

15. What looks random? If a coin is tossed six times and heads(H) or tails(T) are recorded on each toss. Which of the following outcomes is more probable? HTHTTH or TTTHHH We only know that after a long sequence of tosses we should have an equal number of heads and tails but what happens with a short sequence of tosses cannot be predicted.

16. Steaks Joe DiMaggio had a 56 consecutive game hitting streak. Is this remarkable? To investigate, a researcher simulated the performances of every baseball player in every season a total of 10,000 times. In each of those 10,000 simulated “histories of baseball,” they recorded the longest hitting streak. • 42% of the trials of the simulation someone has a hitting streak of at least 56 games in a row • The longest hitting streak in the simulation was 109 games in a row Is Joe DiMaggio’s streak remarkable?

17. Myths About Randomness The myth of short-run regularity Random phenomena are NOT predictable in the short run The myth of the “law of averages” Future outcomes are NOT dependent on the past Law of Large Numbers says that the results will even out in the long run

18. Randomness, Probability, and Simulation • Myths about Randomness The idea of probability seems straightforward. However, there are several myths of chance behavior we must address. The myth of short-run regularity: The idea of probability is that randomness is predictable in the long run. Our intuition tries to tell us random phenomena should also be predictable in the short run. However, probability does not allow us to make short-run predictions. The myth of the “law of averages”: Probability tells us random behavior evens out in the long run. Future outcomes are not affected by past behavior. That is, past outcomes do not influence the likelihood of individual outcomes occurring in the future.

19. Re-cap Questions • Probability is always a value between _____and _____ • Probability answers the question, “What would happen if we did this _____________times?

20. Genetics Problem for Ticket Out Suppose a married man and woman both carry a gene for cystic fibrosis but don’t have the disease themselves. According to the laws of genetics, the probability that their first child will develop cystic fibrosis is 0.25. • Explain what this probability means in context of the situation. • Why doesn’t this probability say that if the couple has 4 children, one of them is guaranteed to get cystic fibrosis?

21. Assignment #1, 7, 9, 11