Each day decisions are based on uncertainty • Should you buy an extended warranty for your new digital camera? • Should you allow 25 min to get school or is 15 min enough? • If an artificial heart has four key parts, how likely is each one to fail? How likely is it that at least one will fail?
Chance experiment – any activity or situation in which there is uncertainty about which of two or more possible outcomes will result. • Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run • Sample space – the collection of all possible outcomes of a chance experiment.
Sample space of chance experiment: • Experiment - whether men and women have different shopping preferences when buying a CD at a music store – classical, rock, country, and “other”. • Tree diagram of sample space:
Event – any collection of outcomes from the sample space of a chance experiment. • Simple event – an event consisting of exactly one outcome
Probability of an event E • P(E) = number of outcomes favorable to E number of outcomes in the sample space ***only when the outcomes of an experiment are equally likely – fair coins or dice, etc.
Probability rules: • Any probability is a number between 0 and 1 • All possible outcomes together must have a probability of 1 • The probability that an event does not occur is 1 minus the probability that the event does occur
Probability Model • A mathematical description of a random phenomenon consisting of a sample space and a way of assigning probabilities to events. • Construct a probability model for a family with 3 children. • All probabilities must add to one • Pg 226 #4.19, 4.20, 4.21
Basic Probability • What is the probability of rolling a 3 with one die? 1/6 • What is the probability of picking a Queen in a deck of cards? 4/52 = 1/13 • What is the probability of picking a heart in a deck of cards? 1/4 • What is the probability of rolling a sum of 8 with two dice? (2,6)(6,2)(3,5)(5,3)(4,4) = 5/36
What is the probability of a family with three children to have all girls? 1/8 • A survey was taken of 10,672 families to determine the number of televisions owned by each. The following results were obtained: Number of Televisions OwnedFrequency 0 72 1 1424 2 2619 3 3227 4 1545 5 1023 6 762 Find the probability of a person having: • Two televisions 2619/10672 • Between one and three televisions, inclusive 7270/10672 = 3635/5336 • Seven televisions 0
Ways of finding # of outcomes: • Counting rule • Permutations – in a line • Arrangement in line with duplicates • Arrangement in circle • Combinations – in a group
Counting rule • If one thing can be done in m ways and if after this is done, something else con be done in n ways, then both things can be done in a total of (m)(n) different ways in the stated order. • A certain model car comes with one of three possible engine sizes and with or without an AM/FM radio. Furthermore, it is equipped with automatic or standard transmission. In how many different ways can a buyer select a car? 3 · 2 · 2 = 12 • There are nine approach roads leading to an airport. Because of heavy traffic, a taxi driver decides to go to the airport by one road and to leave by another road. In how many different ways can this be done? 9 · 8 = 72 • How many different numbers greater than 3000 can be formed from the digits 2, 3, 5, and 9 if no repetitions are allowed? 3 · 3 · 2 · 1 = 18
Permutations • Arrangement of distinct objects in a particular order • Equation: nPr or • In a supermarket there is a long line at the checkout counter. The manager notices this and decides to open an additional checkout counter. Seven people rush over to the new checkout counter. In how many different ways can these seven people line up to be checked out? 7! = 5040 • Susan is an IRS agent. She has made appointments with eight taxpayers to review their 1040 tax forms on May 2 on a first-come, first-served basis. However, due to a computer malfunction, she finds that she has time to meet with only five taxpayers to review their forms. Assuming order counts, in how many different ways can this be done? 8P5 or 8 · 7 · 6 · 5 · 4 = 6720 • Each year movie-goers in a certain city are asked to rank the 5 best movies from among a list of 14 movies. In how many different ways can this be done? 14P5 or 14 · 13 · 12 · 11 · 10 = 240240
The number of different permutations of n things of which p are alike, q are alike, or r are alike • Equation: • How many different ways can we arrange the letters in the word “STATISTICS”.
Arrangement of distinct objects in a circle • Equation: (n – 1)! • How many different ways can four people be arranged in a circle? 3! = 6
Combinations • A selection from a collection of distinct objects where order is not important. • Equation: nCr or • Medical researchers are testing a new drug for treating one form of a neurological disorder. It is decided to select a random group of 18 people and then to select 8 of these people to be given the new drug. The remaining 10 people will be given a placebo. In how many different ways can the 8 subjects be selected? 18C8 = 43758 • There are ten nurses who work on the night shift on the tenth floor of General Hospital. In an effort to save money, the hospital administrator decides to fire four nurses. In how many different ways can the administrator select the four nurses to be fired? 10C4 = 210
Odds • The odds in favor of an event occurring are p to q, where p is the number of favorable outcomes and q is the number of unfavorable outcomes. • Find the odds against rolling a 5 when a single die is rolled once. 5 : 1 • A roulette wheel has 38 slots. One slot is 1, another is 00, and the others are numbered 1 through 36, respectfully. You are placing a bet that the outcome is an odd number. What are the odds of winning? 18 : 38 • What are the odds against winning? 20 : 38
Probability with addition rule • To find the probability of one event or the other to occur, add the probabilities of each one happening • Two dice are rolled. What is the probability that the sum of the dots appearing on both dice together is 9 or 11? 4/36 + 2/36 = 6/36 = 1/6 • One card is drawn from a deck of cards. What is the probability of getting a king or a red card? 4/52 + 26/52 – 2/52 = 28/52
Multiplication rule • If two events are independent events then the probability of both things happening is P(A) · P(B) • Philip, Janet, and Fredric have each applied to different banks for a home equity loan. The probability that Philip’s application is approved is .85. The probability that Janet’s application is approved is .92, and the probability that Fredric’s application is approved is .79. Assuming independence, find the probability that all three applications are approved. (.85)(.92)(.79) = .61778 • From the problem above find the probability that none of the applications are approved. (.15)(.08)(.21) = .00252