Chapter 5: Probability

Transcription

1 Chapter 5: Probability

2 Randomness A random phenomenon is a situation in which we know what outcomes could happen, but we don't know which particular outcome did or will happen However, we can calculate the probability with which each outcome will happen People are not good at identifying truly random samples or random experiments, so we need to rely on outside mechanisms such as coin flips or random number tables

3 Simulating Randomness Flip a coin to generate random 0's and 1's Pick a card to generate random numbers Pick a number out of a hat Use a Random Number table...

4 Simulating Randomness Use a Random Number table Ex. Simulate rolling a die 10 times Pick any line to start, say line 30 to start Go through the numbers, picking numbers 1-6 and ignoring 0,7,8,9. The random numbers are: 5, 4, 5, 3, 4, 6, 2, 5, 3, 1

5 16% of cars fail pollution tests in California We can represent this in many ways: In a bag with 100 chips, 16 are red and represent cars that fail pollution tests, the remaining 84 are black and represent cars that pass the tests. In a bag with 50 marbles, 8 are orange and represent cars that fail pollution tests, the remaining 42 are blue and represent cars that pass the tests. On a random number table numbers represent cars that fail pollution tests and represent cars that pass the tests.

6 Estimating Probabilities via Simulation We would like to estimate the probability that an entire fleet of seven cars would pass using a random number table. We assume each car is independent. Start at row 1 of the random number table and read two digits at a time. If the random number is between 00-15, the car will fail the pollution test. If the number is between 16-99, the car will pass the test. A fleet of cars is comprised of seven cars, i.e. seven 2-digit random numbers. If all seven cars pass the test, record a 1, if not record a 0. Repeat many times and calculate the proportion 1s among the total number of runs, i.e. number of fleets where all cars passed the pollution test.

7 Run 1

8 Run 2

9 Run 3 Did all the cars in run 3 pass? a) Yes? b) No?

10 Run 4

11 Estimation of Probability Based on the simulation results, estimate the probability that an entire fleet of seven cars would pass the pollution test.

12 Notes 4 runs is usually not considered sufficient, for homework use at least 5 runs. You can start at any row you like on the random number table, but you should make sure to note it when you re writing up your simulation scheme. You should not arbitrarily pick numbers from the random number table. Just pick a row and follow across. Otherwise the numbers you re using won t be random (they will instead be your choices).

13 Estimation of Probability Based on the simulation results, estimate the average number of cars that fail the pollution test in a fleet of seven cars. a) 0 b) 1 c) 2 d) 3 e) 4

14 Randomness A random phenomenon is a situation in which we know what outcomes could happen, but we don't know which particular outcome did or will happen However, we can calculate the probability with which each outcome will happen

15 Randomness to Probability Let's say there are 1000 songs in your itunes library. Only 5 of these songs are by Justin Bieber. What is the probability that next time you hit shuffle you get a Justin Bieber song? Prob(JB song) = 5/1000 = 0.005

16 Definitions of Probability A Frequentist: The probability of an event is its long-run relative frequency. Theoretical: We may not be able to predict which song we play each time we hit shuffle, however we know that in the long run 5 out of 1000 songs will be Justin Bieber songs. Empirical: We listen to 100 songs on shuffle and 4 of them are Justin Bieber songs. The empirical probability is 4/100=0.04 A Bayesian: Prior probability of an event is the best guess by the observer of an event's probability. Posterior probability of an event is the probability of an event after collecting some empirical data. It is obtained by updating information from the prior probability with additional data related to the event in question. The prior probability may be subjective, but with enough data, has little impact on the posterior probability.

17 Definitions of Probability Note: In this course we will use the frequentist definition of probability, though we will make some use of Bayesian tools.

18 More Definitions For any random phenomenon, each attempt is called a trial and each trial generates an outcome Sample space is the collection of all possible outcomes of a trial Each time you hit shuffle is a trial. The song that plays as a result of hitting shuffle is an outcome. Sample space is the entire itunes library of 1000 songs. A combination of outcomes is called an event For example playing two Justin Bieber songs in a row in shuffle

19 Sample Space A couple has two kids, what is the sample space for the gender of these kids?

20 Calculating Probabilities If a couple has two kids what is the probability that both are boys? If a couple has two kids, the list of possible scenarios are: BB, GG, BG, GB Since each scenario is equally likely: Prob(BB) = ¼ = 0.25

21 Independence When thinking about what happens with combinations of outcomes, things are simplified if the individual trials are independent. Roughly speaking, this means that the outcome of one trial doesn't influence or change the outcome of another. If the itunes shuffle is truly random then the songs played are independent of each other. In other words, the itunes shuffle is memoryless, it does not say to itself Wait, I just played an Justin Bieber song, I shouldn't play one again Similarly, if the genders of kids a couple has are independent, then the probability of having a boy for a second child doesn't change based on whether or not the first child of the couple was a girl.

22 Definitions Coin toss example Trial: each coin toss Outcome: Heads or Tails Probability: P(H) = 0.5 and P(T) = 0.5 Note: Each time we toss a coin we can't tell which side will come up, however in the long run tails will come up 50% of the time and heads will come up 50% of the time Sample Space: Tossed once: S = {H,T} Tossed twice: S = {HT,TH,HH,TT} Independence: The outcome of one coin toss does not affect the outcome of the next

23 Dice Trial: each die roll Outcome: A die has six sides Probability: With a fair die the probability of getting each side is the same and is 1/6 Sample Space: Rolled once: S = {1, 2, 3, 4, 5, 6} Independence: The outcome of one die roll does not affect the outcome of the next

24 Deck of Cards A deck is made up of 52 cards There are 4 suits: clubs, spades, diamonds, hearts Each suit has 13 cards Within each suit there is one ace, numbers 2-10, a jack, a queen, and a king Clubs and spades are black cards, diamonds and hearts are red cards

25 Law of Large Numbers The Law of Large Numbers (LLN) says that the long-run relative frequency of repeated independent events gets closer and closer to the true relative frequency as the number of trials increases If a coin is tossed many times, the overall percentage of heads should settle down to about 50% as the number of tosses increases This is why we define probability as the long run relativefrequency of an event The common (mis)understanding of the LLN is that random phenomena are supposed to compensate for whatever happened in the past; this is just not true and if also called gambler's fallacy (or law of averages)

26 Law of Large Numbers When tossing a fair coin, if heads comes up on each of the first 10 toss, what do you think the chance is that another head will come up on the next toss? HHHHHHHHHHHHHH? The probability is still 0.5, or there is still a 50% chance that another head will come up on the next toss Prob(H on 11th toss) = Prob(T on 11th toss) = 0.5 The coin is not due for a tails

27 Equally Likely Outcomes Equally likely outcomes have the same probability of happening It's equally likely to get any one of six outcomes from the roll of a fair die It's equally likely to get heads or tails from the toss of a fair coin Not all outcomes are equally likely Its not equally likely to draw an ace as a red card A skilled basketball player has a better than chance of making a free throw

28 Personal Probability WIn everyday speech, when we express a degree of uncertainty without basing it on long-run relative frequencies, we are stating subjective or personal probabilities Personal probabilities don't display the kind of consistency that we will need probabilities to have, so we will stick with formally defined probabilities

29 Impossibility to a Certainty Probabilities must be between 0 and 1, inclusive. A probability of 0 indicates impossibility A probability of 1 indicates certainty Note: If you calculate a probability and get a negative number or a number greater than 1 you are making a mistake, go back and check your work.

30 Something has to happen rule The probability of the seat of all possible outcomes of a trial must be 1 Prob(sample space) = 1 Coin toss: Prob(H) + Prob(T) = =1 Cards: Prob(face card) + Prob(non face card) = 12/ /52 = 52/52 = 1

31 In a class 30% of the students are freshmen, 20% are sophomores, 10% are juniors and the rest are seniors. If we randomly pick a student what is the probability that he or she is a senior? a) 0.3 b) 0.4 c) 0.2 c) 0.1

32 Complement Rule The set of outcomes that are not in the event A is called the complement of A, denoted. The probability of an event occurring is 1 minus the probability that it does not occur.

33 In a survey, 52% of respondents said they are Democrats. What is the probability that a randomly selected respondent from this sample is a Republican? a) greater than 48% b) less than 48% c) equal to 48% d) cannot be computed from the information

34 Disjoint Events Events that have no outcomes in common (and, thus, cannot occur together) are called disjoint (or mutually exclusive). The outcome of a coin toss cannot be a head and a tail A student cannot fail and pass a class A card drawn from a deck cannot be an ace and a queen

35 Addition Rule For two disjoint events A and B, the probability that one OR the other occurs is the sum of the probabilities of the two events. P(A or B) = P(A) + P(B)

36 Do the sum of two disjoint events always add up to one? Yes? No?

37 Multiplication Rule For two independent events A and B, the probability that both A AND B occur is the product of the probabilities of the two events. P(A or B) = P(A) x P(B)

38 In a multiple choice exam there are 5 questions and 4 choices for each question (a, b, c, d). Nancy has not studied for the exam at all, and decided to randomly guess the answers. What is the probability that she gets the first question right? a) 1 c) 0.2 b) 0.25 d) 0.75 What is the probability that she gets the second question right? a) 1 c) 0.25 b) 0.2 d) 0.4

39 Example Guessing at Random What is the probability that the first question she gets right is the 5th question? Prob(Wrong Wrong Wrong Wrong Right) = 0.75 x 0.75 x 0.75 x 0.75 x 0.25 = What is the probability that she gets all the questions right? a) 1.25 c) b) d) 0

40 Example Guessing at Random What is the probability that the first question she gets at least one question right? If there are 5 questions, the possible number of questions she gets right are: S = {0,1,2,3,4,5} We are interested in instances where she gets at least 1 question right: S = {0,1,2,3,4,5} So we can divide up the sample space into two categories: S = {0, at least 1}

41 Example Guessing at Random

42 Independent vs. Disjoint Can two events be independent and disjoint? Remember: Independence means that the outcome of one trial doesn't influence the outcome of another and disjoint means that two events cannot occur together. If we know that the outcome of a coin toss is heads, then we know that it is not tails. So whether or not the outcome is tails depends on whether or not the outcome is heads. Therefore these two events are disjoint and dependent.

43 Independent vs. Disjoint Disjoint events cannot be independent Since we know that disjoint events have no outcomes in common, knowing that one occurred means the other didn't happen. Thus, the probability of the one event occurring changed based on our knowledge that the other occurred. Therefore the two events are not independent. Non-disjoint events may or may not be independent Just because two events can occur at the same time does not mean they need to also be dependent. It is possible that two randomly selected students get an A in a class, but how they do in the class may have nothing to do with each other. However if you and a close friend of yours that you study with both get an A in a class, how well the two of you do in that class are possibly dependent on each other.

44 Properties of Probability - Summary When thinking about what happens with combinations of outcomes, things are simplified if the individual trials are independent. Roughly speaking, this means that the outcome of one trial doesn't influence or change the outcome of another. If the itunes shuffle is truly random then the songs played are independent of each other. IIn other words, the itunes shuffle is memoryless, it does not say to itself Wait, I just played an Justin Bieber song, I shouldn't play one again

45 Example Absence from School A middle school estimates that 20% of its students miss one day of school per semester due to sickness, 13% miss two days, and 5% miss three or more days. What is the probability that a student chosen at random doesn't miss any days of school due to sickness? Prob(no misses) = 1 - ( ) = 0.62

46 Example Absence from School A middle school estimates that 20% of its students miss one day of school per semester due to sickness, 13% miss two days, and 5% miss three or more days. What is the probability that a student chosen at random doesn't miss any days of school due to sickness? a) 0.38 c) 0.18 b) 0.33 d) 0.62

47 Example Absence from School A middle school estimates that 20% of its students miss one day of school per semester due to sickness, 13% miss two days, and 5% miss three or more days. What is the probability that a student chosen at random misses no more than one day? Prob(at most 1 miss) = Prob(no misses) + Prob(1 miss) = = 0.82

48 Example Absence from School A middle school estimates that 20% of its students miss one day of school per semester due to sickness, 13% miss two days, and 5% miss three or more days. If a parent has two kids at this middle school, what is the probability that neither will miss any school? Prob(neither miss any) = Prob(no miss) x Prob(no miss) 0.62 x 0.62 =

49 Example Absence from School We used the multiplication rule to calculate the probability in the previous slide. a) What must be true about these kids to make that approach valid? b) Do you think this assumption is reasonable?

50 Example Absence from School A middle school estimates that 20% of its students miss one day of school per semester due to sickness, 13% miss two days, and 5% miss three or more days. If a parent has two kids at this middle school, assuming that the kids are independent with respect to missing school, what is the probability that both will miss some school, i.e. at least one day? Choose the closest answer. a) 0.14 c) 0.38 b) 0.24 d) 0.76

51 Example Absence from School A middle school estimates that 20% of its students miss one day of school per semester due to sickness, 13% miss two days, and 5% miss three or more days. If a parent has two kids at this middle school, assuming that the kids are independent with respect to missing school, what is the probability that one kid misses some school and the other doesn't miss any?

52 General Addition Rule

53 Example Second Language In a class where everyone's native language is English, 70% of students speak Spanish, 45% speak French as a second language and 20% speak both. Assume that there are no students who speak another second language. What is the probability that a randomly selected students speaks Spanish or French? Prob(Spanish or French) = Prob(Spanish) + Prob(French) - Prob(Spanish and French) = = 0.95

54 Picturing Probabilities The most common kind of picture we will make to picture probabilities is called Venn Diagram.

55 Example Second Language In a class where everyone's native language is English, 70% of students speak Spanish, 45% speak French as a second language and 20% speak both. Assume that there are no students who speak another second language.

56 Conditional Probability

57 Example Second Language In a class where everyone's native language is English, 70% of students speak Spanish, 45% speak French as a second language and 20% speak both. Assume that there are no students who speak another second language. What is the probability that a randomly selected student speaks French given that they speak Spanish?

58 Example Second Language In a class where everyone's native language is English, 70% of students speak Spanish, 45% speak French as a second language and 20% speak both. Assume that there are no students who speak another second language. What is the probability that a randomly selected student speaks Spanish given that they speak French? a) 0.45 c) 2.25 b) 0.44 d) 0.05

59 Example Burgers in LA A survey conducted in 2010 by Survey USA for KABC-TV Los Angeles asked 500 Los Angeles residents What is the best hamburger place in Southern California? Five Guys Burgers? In-N-Out Burger? Fat Burger? Tommys Hamburgers? Umami Burger? Or somewhere else? Consider the results of this survey, shown in the table below, in each of the following questions.

60 Example Burgers in LA Refer to the table on the previous slide: What is the probability that a randomly chosen male likes In-N-Out the best? What is the probability that a randomly chosen female likes In-N-Out the best? What is the probability that a man AND a woman who are dating both like In-N-Out the best?

61 Example Burgers in LA What is the probability that a randomly chosen person likes Umami best or that the person is a female? a) c) b) d) 0.516

62 General Multiplication Rule

63 Checking for independence using conditional probabilities

64 Example Second Language In a class where everyone's native language is English, 70% of students speak Spanish, 45% speak French as a second language and 20% speak both. Assume that there are no students who speak another second language. Are the variables speaking French and speaking Spanish independent?

65 Independent vs disjoint - recap If A and B are disjoint: Prob(A and B) = 0 A and B are not independent If A and B are not disjoint: Prob(A and B) > 0 A and B may or may not be independent

66 Example An online survey shows that 25% of bloggers own a DSLR camera, 90% own a point-and-shoot camera, and 20% own both types of cameras. Are owning a DSLR and point-andshoot camera disjoint?

67 Example An online survey shows that 25% of bloggers own a DSLR camera, 90% own a point-and-shoot camera, and 20% own both types of cameras. Are the variables owning a DSLR and owning a point-and-shoot camera independent?