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ARE YOU FOR OR AGAINST A MILITARY STRIKE AGAINST SYRIA? 1. 2. 3. 4. For Against It’s a lot more complicated Where’s Syria? 17% 17% 17% 17% 17% 17% Slide 1- 1 1 2 3 4 5 6 UPCOMING IN CLASS Wednesday’s Quiz 2, covers HW3 of the material learned in class. (9/11) Sunday HW4 (9/15) Part 2 of the data project due next Monday (9/16) CHAPTER 12 Probability Properties: Addition and Multiplication under Disjoint and Independence DEALING WITH RANDOM PHENOMENA 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. In general, each occasion upon which we observe a random phenomenon is called a trial. At each trial, we note the value of the random phenomenon, and call it an outcome. When we combine outcomes, the resulting combination is an event. The collection of all possible outcomes is called the sample space. Slide 1- 4 WHICH OF THE FOLLOWING IS THE SAMPLE SPACE FOR RECORDING THE ORDER OF HEADS AND TAILS WHEN TOSSING 3 COINS? 25% 1. 2. 3. 4. 25% 25% 25% {HHT, HTT, HTH, TTT} {HHH,HHT,HTH, HTT, THT,TTH, THH, TTT} {H,T, HH, HT, TH TT} {H,T} Slide 1- 5 1 2 3 4 A FAMILY HAS 5 CHILDREN. WHAT IS THE SAMPLE SPACE FOR THE NUMBER OF BOYS IN THE FAMILY? 1. 2. 3. 4. {1,2,3,4,5} {0,1,2,3,4,5} {0,1,2,3,4} {1,2,3,4} 25% 25% 25% 25% Slide 1- 6 1 2 3 4 DURING A BASKETBALL GAME A PLAYER TAKES A SHOT. WHAT IS THE SAMPLE SPACE FOR THE NUMBER OF POINTS? 1. 2. 3. 4. {0, 1, 2, 3} {-1, 1, 2, 3} {1, 2, 3} {0, 1, 2} 25% 25% 25% 25% Slide 1- 7 1 2 3 4 MODELING PROBABILITY WITH RELATIVE FREQUENCY The probability of an event is the number of outcomes in the event divided by the total number of possible outcomes. P(A) = # of outcomes in A # of possible outcomes THE LAW OF LARGE NUMBERS (CONT.) The Law of Large Numbers (LLN) says that the longrun relative frequency of repeated independent events gets closer and closer to a single value (i.e. true probability). 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. For example, coin flips are independent. Because this definition is based on repeatedly observing the event’s outcome, this definition of probability is often called empirical probability. Slide 1- 9 EVENT SIMULATOR Coin Toss LLN Experiment (Redline) http://socr.ucla.edu/htmls/SOCR_Experiments.ht ml Slide 1- 10 FORMAL PROBABILITY 1. Two requirements for a probability: A probability is a number between 0 and 1. For any event A, 0 ≤ P(A) ≤ 1. Slide 1- 11 FORMAL PROBABILITY (CONT.) 2. Probability Assignment Rule: The probability of the set of all possible outcomes of a trial must be 1. P(S) = 1 (S represents the set of all possible outcomes.) Slide 1- 12 ENTER QUESTION TEXT... 25% 1. 2. 3. 4. 25% 25% 25% A,B,C,D,E A,B,C A,B,D,E A,B,C,D Slide 1- 13 1 2 3 4 THE 68-95-99.7 RULE (CONT.) The following shows what the 68-95-99.7 Rule tells us: Slide 1- 14 FORMAL PROBABILITY (CONT.) 3. Complement Rule: The set of outcomes that are not in the event A is called the complement of A, denoted AC. The probability of an event occurring is 1 minus the probability that it doesn’t occur: P(A) = 1 – P(AC) Slide 1- 15 FORMAL PROBABILITY (CONT.) 4. Addition Rule: Events that have no outcomes in common (and, thus, cannot occur together) are called disjoint (or mutually exclusive). Slide 1- 16 WHICH OF THE FOLLOWING EVENTS ARE DISJOINT? 1. 2. 3. 4. Being black and female Being republican and female Being female and an athlete Being female and having a Y chromosome 25% 25% 25% 25% Slide 1- 17 1 2 3 4 FORMAL PROBABILITY (CONT.) 4. Addition Rule (cont.): 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), provided that A and B are disjoint. Slide 1- 18 ENTER QUESTION TEXT... 25% 1. 2. 3. 4. 25% 25% 25% =513/1121 =511/1121 =62/1121 =35/1121 Slide 1- 19 1 2 3 4 ENTER QUESTION TEXT... 25% 1. 2. 3. 4. 25% 25% 25% =513/1121 =511/1121 =(62+35)/1121 =35/62 Slide 1- 20 1 2 3 4 FORMAL PROBABILITY 5. Multiplication Rule (cont.): 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 and B) = P(A) x P(B), provided that A and B are independent. FORMAL PROBABILITY - NOTATION Notation alert: In this text we use the notation P(A or B) and P(A and B). In other situations, you might see the following: P(A B) instead of P(A or B) P(A B) instead of P(A and B) Slide 1- 22 ENTER QUESTION TEXT... 25% 1. 2. 3. 4. 25% 25% 25% 1/3 3 1/6 + 1/6 +1/6 1/6 * 1/6 * 1/6 Slide 1- 23 1 2 3 4 ENTER QUESTION TEXT... 25% 1. 2. 3. 4. 25% 25% 25% 1/2 2 1/2 + 1/2 +1/2 1/2 * 1/2 * 1/2 Slide 1- 24 1 2 3 4 ENTER QUESTION TEXT... 25% 1. 2. 3. 4. 25% 25% 25% 2/6*2/6*2/6 2/6+2/6+2/6 1- 2/6+2/6+2/6 4/6 * 4/6 * 4/6 Slide 1- 25 1 2 3 4 ENTER QUESTION TEXT... 25% 1. 2. 3. 4. 25% 25% 25% 1/6*1/6*1/6 1-1/6*1/6*1/6 1-5/6*5/6*5/6 5/6 * 5/6 * 5/6 Slide 1- 26 1 2 3 4 ENTER QUESTION TEXT... 25% 1. 2. 3. 4. 25% 25% 25% 1/6*1/6*1/6 1-1/6*1/6*1/6 1-5/6*5/6*5/6 5/6 * 5/6 * 5/6 Slide 1- 27 1 2 3 4 WHAT CAN GO WRONG? (CONT.) Don’t multiply probabilities of events if they’re not independent. The multiplication of probabilities of events that are not independent is one of the most common errors people make in dealing with probabilities. Don’t confuse disjoint and independent—disjoint events can’t be independent. Slide 1- 28 CAR PROBLEM A consumer organization estimates that over a 1year period 16% of cars will need to be repaired once, 7% will need repairs twice, and 1% will require three or more repairs. Suppose you own two cars. Slide 1- 29 WHAT IS THE PROBABILITY THAT BOTH CARS WILL NEED REPAIRED THIS YEAR? 1. 2. 3. 4. .76 .24 .5776 .0576 25% 25% 25% 25% Slide 1- 30 1. 2. 3. 4. WHAT IS THE PROBABILITY THAT NEITHER CAR WILL NEED REPAIRED THIS YEAR? 1. 2. 3. 4. .76 .24 .5776 .0576 25% 25% 25% 25% Slide 1- 31 1. 2. 3. 4. WHAT IS THE PROBABILITY THAT AT LEAST ONE CAR WILL NEED REPAIRED THIS YEAR? 1. 2. 3. 4. .76 .24 .5776 .4224 25% 25% 25% 25% Slide 1- 32 1. 2. 3. 4. LOTTERY PROBLEM On September 11, 2002, a particular state lottery’s daily number came up 9-1-1. Assume that no more than one digit is used to represent the first nine months. Slide 1- 33 WHAT IS THE PROBABILITY THAT THE WINNING THREE NUMBERS MATCH THE DATE ON ANY GIVEN DAY THAN CAN BE REPRESENTED BY A THREE DIGIT NUMBER? 25% 25% 25% 1. 2. 3. 4. 25% 0.000 0.001 0.273 1 Slide 1- 34 1 2. 3. 4 WHAT’S THE PROBABILITY THAT THE WINNING THREE NUMBER MATCH THE DATE ON ANY GIVEN DAY THAT CAN BE REPRESENTED BY A 4DIGIT NUMBER? 25% 25% 25% 25% 1. 2. 3. 4. 0.000 0.001 0.273 1 Slide 1- 35 1 2. 3. 4 WHAT’S THE PROBABILITY THAT A WHOLE YEAR PASSES WITHOUT THE LOTTERY NUMBER MATCHING THE DAY? 1. 2. 3. 4. 0.001 0.694 0.761 0.999 25% 25% 25% 25% Slide 1- 36 1. 2. 3. 4. UPCOMING IN CLASS Wednesday’s Quiz 2, covers HW3 of the material learned in class. (9/11) Sunday HW4 (9/15) Part 2 of the data project due next Monday (9/16)