Series Time Plot: a simple graph of data collected over time that can be invaluable in identifying trends or patterns that might be of interest. Life Expectancy over Time Year Life Expectancy over Time 1940 62.9 1950 68.2 1960 69.7 1970 70.8 1980 73.7 1990 75.4 2000 76.9
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5.1 How Can Probability Quantify Randomness? Random phenomenon We cannot predict the next outcome, but a regular and predictable pattern emerges in the long run. This pattern follows the laws of probability. Random does NOT mean haphazard it is a kind of order that emerges in the long run. Example: Studying the concept of probability when tossing a fair coin. a) b) What is the probability of getting a head if we toss a fair coin? Flip the coin once. What is the result? Tails or Heads? ______________ Proportion of heads obtained in all your tosses so far: ______________ Is this close to the probability of heads you expected? Why? Flip the coin nine more times. What are the results? _______________________ Proportion of heads obtained in all your tosses so far: _______________ Is this close to the probability of heads you expected? ______Why? Imagine flipping the coin a thousand times. What will the results be? ______________________ Should the proportion of heads obtained in all your tosses be closer to the probability of heads you expected? Why? ___________________________________
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Probability Quantifies Long-Run Randomness Probability of an outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of independent trials. Probability is a long-term relative frequency. Computer simulations are useful for "pretending" we are performing an experiment a very large number of times. Independent trials: the outcome of any one trial is not affected by the outcome of any other. Why do we need probability? Probability is used to describe games of chance, the flow of traffic through a highway or the Internet, the spread of diseases or rumors, the rate of return of investments, etc. 5.2 How Can We Find Probabilities? Probability of an event E: denoted P(E),
P(E)
number of outcomes favorable to E number of outcomes in the sample space
Note: This method for calculating probabilities is appropriate only when the outcomes of an experiment are equally likely. Rolling Dice: P(rolling a 6): P(rolling an even number): Chance experiment: any activity or situation in which there is uncertainty about which of two or more possible outcomes will result. Sample space: the set of all possible outcomes. Event: an outcome or group of outcomes, a subset of the sample space. Probabilities are between 0 and 1. The sum of all outcome probabilities must equal 1. CD's: Consider a chance experiment designed to see whether men and women have different shopping preferences when buying a CD at a music store. Four types of music are sold at the store: classical, rock, country, and "other". For this experiment, a randomly selected shopper will be asked what type of CD has been purchased, and the investigator will note the shopper's gender. Sample Space: List: 1. 2. 3. 4. Set Notation:
5. 6. 7. 8.
Tree Diagram:
Simple event: an event consisting of exactly one outcome. CD's revisited: Events: Simple Events: Classical: Male: Examples: Write down the sample space and the probability of each event. a) b) Toss a fair coin, record the number of heads. Sample space = P(head) = Toss two coins and record number of heads. Sample space = P(0) = P(1) = P(2) =
P(tail) =
Cards: Consider the experiment consisting of randomly picking a card from an ordinary deck of playing cards (52 card deck). Let A stand for the event that the card chosen is a King. S = sample space = A= P(A) = Dice: Consider the experiment consisting of rolling two fair dice and observing the sum of the up faces. Let E stand for the event that the sum is 7 and F stand for the event that the sum is 11. S= E= P(E) = F= P(F) = Basic Rules for Finding Probabilities about a Pair of events Complement Rule: The probability that an event does not occur is 1 minus the probability that it does occur. For an event A, the rest of the sample space that is not in A is the complement, A c. The P(Ac) = 1 P(A). Disjoint Events: If two events A and B have no outcomes in common they are called disjoint events. Then the probability that one or the other occurs is the sum of their individual probabilities. That is (A or B) = P(A) + P(B). Intersection of two events A and B is the overlap of the two events that are in both A and B. For disjoint events P(A and B) = P(A B) = 0. Union of two events A and B, consists of all events that are in A or B or both. In general P(A or B) = P(A) + P(B) P(A B). Independence: If two events are independent, knowledge about one event tells us nothing about the other event. Multiplication Rule: If A and B are independent, then P(A and B) = P(A) P(B).
Venn Diagram: an informal picture that is used to identify relationships.
The collection of all possible outcomes of a chance experiment are represented as the interior of a rectangle. The rectangle represents sample the space and shaded area represents the event A. Forming New Events: Let A and B denote two events: Not A (Complement of A): consists of all experimental outcomes that are not in event A.
A or B (The union of A and B): consists of all experimental outcomes that are in at least one of the two events, that is, in A or in B or in both of these. Denoted A B.
A and B (The intersection of A and B): consists of all experimental outcomes that are in both the event A and the event B. Denoted AB.
Disjoint (Mutually Exclusive): two events that have no events in common.
Some illustrations
A C
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Cash or Credit: Customers at a certain department store pay for purchases with either cash or one of four types of credit card. Store records, kept for a long period of time, indicate that 30% of all purchases involve cash, 25% are made with the store's own credit card, 18% with MasterCard (MC), 15% with Visa (V), and the remaining 12% with American Express (AE). The following table displays the probabilities of the simple events for the chance experiment in which the mode of payment for a randomly selected transaction is observed: Simple Event: O1 (Cash) Probability: .30 O2 (Store) .25 O3 (MC) .18 O4 (V) .15 O5 (AE) .12
E is the event a randomly selected purchase is made with a nationally distributed credit card. (MC, V, AE) P(E) = P(not E) =
Turning Directions: an observer stands at the bottom of a freeway off-ramp and records and turning direction (L = left or R = right) of each of three successive vehicles. Sample space: A = the event that exactly one of the cars turns right = B = the event that at most one of the cars turn right = C = the event that all cars turn in the same direction = not C = A or C = B and C = A and C = Example: Grades on a very large statistics class are given according to the following distribution: A 15% a) b) c) d) B 35% C 30% D 16% F ?
What is the probability that a student gets an F in the class? What is the probability that a student gets better than a C? What is the probability that a students gets a C or better? What is the probability that two randomly selected students (independent of each other) both get an A in the class?
Example: Deal cards from a standard deck (13 each hearts, diamonds, spades, clubs). Can we use the multiplication rule to find the probabilities of different hands? Hint: Are all cards independent? 5.3 Conditional Probability: What Is the Probability of A, Given B? Conditional Probability: Let E and F be two events with P(F) > 0. The conditional probability of the event E given that the event F has occurred, denoted by P(E|F), is
P(E | F) P(E F) P(F)
Job Satisfaction: A survey of job satisfaction2 of teachers was taken, giving the following results
Job Satisfaction Satisfied Unsatisfied Total College 74 43 117 High School 224 171 395 Elementary 126 140 266 Total 424 354 778
L E V E L
Divide all cells by the total number surveyed: Satisfied Unsatisfied Total College High School Elementary Total For convenience, let C stand for the event that the teacher teaches college, S stand for the teacher being satisfied and so on. Let's look at some probabilities and what they mean. P(C) = P(S) = P(C S) = P(C|S) = Example: Do male college students follow their school's team more closely than females? The following data was collected in class on a Monday morning, after a particularly exciting and important basketball game. The question asked was: Did you watch the game on TV last night? Whole Game 10 Male Female 21 Total 31 Part of the Game 12 24 36 None of It 4 30 34 Total 26 75 101
Find the conditional probability of a student having watched none of the game, given that the student was a female.
Independent Events and Conditional Probability Recall two events A and B are independent if knowledge about one event tells us nothing about the other event. With this definition of conditional probability we can write: A and B are independent if: P(A|B) = P(A) If A and B are independent then: P(A and B) = P(A B) = P(A) P(B) Job Satisfaction revisited: P(C) = P(C|S) = Does the P(C) = P(C|S)? Are C and S independent? TV and VCR: Consider the person who purchases from two different manufacturers a TV and a VCR. Suppose we define the events A and B by A = event the TV doesn't work properly B = event the VCR doesn't work properly Suppose P(A) = 0.01 and P(B) = 0.02. If we assume the events A and B are independent then P(A B) =
5.4 Applying the Probability Rules Example: Random drug testing of air traffic controllers. Suppose the probability that an air traffic controller uses illegal drugs is 7/1000 (.007). No test is perfect, but the drug test used here correctly identifies drug users with probability of .96. It also correctly identifies non-drug users with a probability of .93. a) Find the probability of a test giving a positive result. Test Positive Test Negative Total Drug User Non-drug user Total b) Find the probability that a person uses drugs given that their test result was positive.
c)
If 100 non-drug users are tested, find the probability that at least one of them gets a (mistakenly) positive result.