1. Copyright © 2014 Pearson Education, Inc. All rights reserved Chapter 5 Modeling Variation with Probability

2. 5 - 2 Copyright © 2014 Pearson Education, Inc. All rights reserved Learning Objectives  Understand that humans can’t reliably create random numbers or sequences.  Understand that a probability is a long-term relative frequency.  Know the difference between empirical and theoretical probabilities— and know how to calculate them.

3. 5 - 3 Copyright © 2014 Pearson Education, Inc. All rights reserved Learning Objectives Continued  Be able to determine whether two events are independent or associated and understand the implications of making incorrect assumptions about independent events.  Understand that the Law of Large Numbers allows us to use empirical probabilities to estimate and test theoretical probabilities.  Know how to design a simulation to estimate empirical probabilities.

4. Copyright © 2014 Pearson Education, Inc. All rights reserved 5.1 What Is Randomness?

5. 5 - 5 Copyright © 2014 Pearson Education, Inc. All rights reserved Randomness  If numbers are chosen at random, then no predictable pattern occurs and no digit is more likely to appear more often than another.  In general, outcomes occur at random if every outcome is just as likely to appear as any other outcome and no predictable pattern of outcomes occurs.

6. 5 - 6 Copyright © 2014 Pearson Education, Inc. All rights reserved Psychology and Randomness  Pick a “random” number between 1 and 20.  This may seem random, but due to cognitive bias some numbers, e.g. 7, are more likely than others. Odd numbers and especially prime numbers “feel” more random.

7. 5 - 7 Copyright © 2014 Pearson Education, Inc. All rights reserved Using a Random Number Table to Simulate Rolling a Die 10 Times 1. Pick a line, say 30, on the table to begin. 2. Select numbers in order disregarding 0,7,8,9. 3. The “random” numbers are 5,4,5,3,4,6,2,5,3

8. 5 - 8 Copyright © 2014 Pearson Education, Inc. All rights reserved Troubles With Tables and Computers  The random number table only has a finite list. If it is used many times, it will not be random at all.  Computers involve a random seed, typically given by the time it is clicked to the nearest millisecond.

9. 5 - 9 Copyright © 2014 Pearson Education, Inc. All rights reserved Other Physical Techniques  Flip a coin to generate random 0’s and 1’s.  Pick a card to generate random numbers.  Roll a die to generate numbers from 1 to 6.  Pick a number out of a hat.  Warning: Skilled magicians can manipulate coins, cards, dice, and hats to select a value of their choice.

10. 5 - 10 Copyright © 2014 Pearson Education, Inc. All rights reserved Empirical and Theoretical Probabilities  Probability measures the proportion or percent of the time that a random event occurs.  Theoretical probabilities are long run relative frequencies based on theory.  P(Heads) = 0.5  Empirical probabilities are short run relative frequencies base on an experiment.  A coin was tossed 50 times and landed on heads 22 times. The empirical probability is 22/50 = 0.44.

11. 5 - 11 Copyright © 2014 Pearson Education, Inc. All rights reserved Using Theoretical and Empirical Probabilities  Use Theoretical Probabilities when we can mathematically determine them.  Dice, Cards, Coins, Genetics, etc.  Use Empirical Probabilities when they cannot be mathematically determined. This is done by sampling.  Weather, Politics, Business Success, etc.

12. Copyright © 2014 Pearson Education, Inc. All rights reserved 5.2 Finding Theoretical Probabilities

13. 5 - 13 Copyright © 2014 Pearson Education, Inc. All rights reserved Probability Properties  0 ≤ P(A) ≤ 1  There can’t be a negative chance or more than a 100% chance of something occurring.  P(A c ) = 1 – P(A)  A c the complement of A means that A does not occur.  If there is a 25% chance of winning, then there is a 75% chance of not winning.

14. 5 - 14 Copyright © 2014 Pearson Education, Inc. All rights reserved Probability for Equally Likely Events  If all events are equally likely, then  Example: Find the probability of picking an Ace from a 52 card deck. 

15. 5 - 15 Copyright © 2014 Pearson Education, Inc. All rights reserved Sum of Dice  Roll 2 dice. Find P(Sum = 6). 123456 1X 2X 3X 4X 5X 6  Sum of 6: 5  Possible Rolls: 36  P(Sum = 6) =

16. 5 - 16 Copyright © 2014 Pearson Education, Inc. All rights reserved AND  The word “And” in Probability means both must occur.  Example: If you roll a die, find the probability that it is even and less than 5.  Solution: The die rolls that are both even and less than 5 are: 2, 4.

17. 5 - 17 Copyright © 2014 Pearson Education, Inc. All rights reserved OR  The word “OR” in probability means at least one of the events must occur.  Example: Find the probability of picking a Spade Or a King from a 52 card deck.  Solution: There are 13 spades in the deck. There are 3 kings that are not spades. Thus, there are 16 cards that are a spade or a king.

18. 5 - 18 Copyright © 2014 Pearson Education, Inc. All rights reserved Mutually Exclusive Events  Two events are called Mutually Exclusive if they cannot both occur.  If A and B are Mutually Exclusive then P(A AND B) = 0  Example: A person is selected at random. Let A be the event that the person is a registered Democrat and let B be the event that the person is a registered Republican. Then A and B are mutually exclusive events.

19. 5 - 19 Copyright © 2014 Pearson Education, Inc. All rights reserved Some Probability Rules 1. P(A c ) = 1 – P(A) 2. P(A OR B) = P(A) + P(B) – P(A AND B) 3. Mutually Exclusive: a. P(A OR B) = P(A) + P(B) b. P(A AND B) = 0

20. 5 - 20 Copyright © 2014 Pearson Education, Inc. All rights reserved Venn Diagrams  A Venn Diagram is a chart that organizes outcomes.  P(Hat AND Glasses) = 2/6  P(Not Hat) = 1 – P(Hat) = 1 – 3/6 = 1/2

21. 5 - 21 Copyright © 2014 Pearson Education, Inc. All rights reserved Tables and Probability  Take a bus.  55/278  Be male and take a bike  14/278 CarBikeWalkBusTotal Male75141223124 Female9072532154 Total165213755278 Transportation to Class Find the probability that a randomly selected student will:  Be female or walk to class.  154/278 + 37/278 – 25/278 = 166/278

22. Copyright © 2014 Pearson Education, Inc. All rights reserved 5.3 Associations in Categorical Variables

23. 5 - 23 Copyright © 2014 Pearson Education, Inc. All rights reserved Conditional Probability  Conditional Probability is the probability of an event occurring given some additional knowledge.  Find the probability that a person will vote for a tax cut given that the person is Republican.  Find the probability that student who is a psychology major is also a vegetarian.

24. 5 - 24 Copyright © 2014 Pearson Education, Inc. All rights reserved Tables and Conditional Probability  P(Bus|Female) = Probability of riding the bus given that the person is female.  32/154  P(Bus AND Female) = 32/278 CarBikeWalkBusTotal Male75141223124 Female9072532154 Total165213755278

25. 5 - 25 Copyright © 2014 Pearson Education, Inc. All rights reserved Determining a Probability Statement  Let A be the event that a person is left handed and B be the event that the person is over 30 year old. Write symbolically: The probability that:  a left handed person will be over 30.  P(B|A)  a person is a lefty and who is over 30.  P(A AND B)  a person over 30 years old is a righty.  P(A C |B)

26. 5 - 26 Copyright © 2014 Pearson Education, Inc. All rights reserved The Formula for a Conditional Probability  Use this formula when explicitly given the probabilities or percents.  You do not need to use this formula when given a contingency table.

27. 5 - 27 Copyright © 2014 Pearson Education, Inc. All rights reserved Variations of the Formula

28. 5 - 28 Copyright © 2014 Pearson Education, Inc. All rights reserved Conditional Probabilities  By 2020, 2% of Americans will be Senior Citizens living in poverty. 17% of all Americans will be Senior Citizens in 2020. What percent of all Senior Citizens will be living in poverty?  A → Senior Citizen, B → Living in Poverty  P(A AND B) = 0.02, P(A) = 0.17  P(B|A) = 0.02/0.17 ≈ 0.12  In 2020, about 12% of all Senior Citizens will be living in poverty.

29. 5 - 29 Copyright © 2014 Pearson Education, Inc. All rights reserved Independent Events  Events A and B are called independent if  P(A|B) = P(A) or equivalently P(A AND B) = P(A)P(B)  Intuitively, events are independent if knowledge of B does not change the probability of A occurring.

30. 5 - 30 Copyright © 2014 Pearson Education, Inc. All rights reserved Determining Independence  Two 6-Sided dice are rolled. Let A be the event that the dice sum to 7 and B be the event that the first die lands on a 4. Are A and B independent?  (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)  P(A) = 6/36  P(A|B) = 1/6  P(A|B) = P(A)  Yes, the events are independent.

31. Copyright © 2014 Pearson Education, Inc. All rights reserved 5.4 The Law of Large Numbers

32. 5 - 32 Copyright © 2014 Pearson Education, Inc. All rights reserved Empirical Probability Example  The table to the right shows the result of tossing a coin 10 times.  Empirical Probabilities of heads:  After first toss: P(H) = 1/1 = 1.00  After second toss: P(H) = 2/2 = 1.00  After third toss: P(H) = 3/3 = 1.00  After fourth toss: P(H) = 3/4 = 0.75  After tenth toss: P(H) = ?  Empirical probability is not the same as theoretical probability.

33. 5 - 33 Copyright © 2014 Pearson Education, Inc. All rights reserved The Law of Large Numbers  The Law of Large Numbers states that if an experiment with a random outcome is repeated a large number of times, the empirical probability of an event is likely to be close to the true probability. The larger the number of repetitions, the closer together these probabilities are likely to be.

34. 5 - 34 Copyright © 2014 Pearson Education, Inc. All rights reserved The Law of Large Numbers Examples  If you flip a fair coin one million times, it is likely to land on heads close to half the time.  If you randomly survey 50,000 Americans asking them if they know what the capitol of Alabama is, the proportion from the survey who do know will be very close to the proportion of all Americans who know.

35. 5 - 35 Copyright © 2014 Pearson Education, Inc. All rights reserved Warnings About the Law of Large Numbers  If the theoretical probability if far from 0.5, use a very large number of trials for the Empirical Probability to be close.  If you flip a fair coin five times and it lands on heads all five times, this does not mean that it will land on tails the next five times to compensate.

36. 5 - 36 Copyright © 2014 Pearson Education, Inc. All rights reserved Law of Large Numbers Does Not Say That Streaks Cannot Occur  If the first five tosses of a coin all land heads, this does not violate the Law of Large Numbers.  If you just watched a fair die rolled 20 times without seeing a 2, this does not mean that a 2 is due on the next toss.