Probability: Living with the Odds 6D Discussion Paragraph 1 web 50. Vital Significance 51. Recent Polls 52. Genetically Modified Foods 1 world 53. Statistical Significance 54. Significant Experiment? 55. Margin of Error 56. Confidence Intervals 57. Hypothesis Testing Copyright © 2011 Pearson Education, Inc.

Fundamentals of Probability Unit 7A Fundamentals of Probability Copyright © 2011 Pearson Education, Inc.

Lotteries Lotteries are big business and a major revenue source for many governments. By 2009, all but seven states in the US had some type of lottery, generating a national total of more than $52 billion in sales. About 1/3 of that money ($17 billion) ends up as state revenue, with the rest going to prizes and expenses. Lotteries present many lessons in probability, and lottery stats can fuel great debate over whether lotteries are an appropriate way for governments to generate revenue. Copyright © 2011 Pearson Education, Inc.

Definitions Outcomes are the most basic possible results of observations or experiments. An event consists of one or more outcomes that share a property of interest. Copyright © 2011 Pearson Education, Inc.

Expressing Probability The probability of an event, expressed as P(event), is always between 0 and 1 (inclusive). A probability of 0 means the event is impossible and a probability of 1 means the event is certain. 1 0.5 Certain Likely Unlikely 50-50 Chance Impossible 0 ≤ P(A) ≤ 1 Meteorology offers a rich source of ideas here. Would you take an umbrella to work if the chance for rain is at 10%? How about 90%? Copyright © 2011 Pearson Education, Inc.

Theoretical Method for Equally Likely Outcomes Step 1: Count the total number of possible outcomes. Step 2: Among all the possible outcomes, count the number of ways the event of interest, A, can occur. Step 3: Determine the probability, P(A). Copyright © 2011 Pearson Education, Inc.

Outcomes and Events Assuming equal chance of having a boy or girl at birth, what is the probability of having two girls and two boys in a family of four children? Of the 16 possible outcomes, 6 have the event two girls and two boys. P(2 girls) = 6/16 = 0.357 One of the challenges many students have with probability is knowing when to trust intuition and when to back away from it. Many students would answer the question with a 50%. This problem is a classic example of the importance of looking at the total number of outcomes. Copyright © 2011 Pearson Education, Inc.

Playing Card Probabilities CN (1) There are 52 cards in a standard deck. There are four suits, hearts, diamonds, spades and clubs. Each suit has card for the numbers 2 through 10 plus a jack, queen, king, and ace (for a total of 13 cards in each suit). Notice that 2 suits are red and 2 suits are black. 1. If you draw one card from a standard deck, what is the probability that it is a spade? Copyright © 2011 Pearson Education, Inc.

Guessing Birthdays CN (2) You select a person at random from a large conference group. 2. What is the probability that the person has a birthday in July? Assume 365 days a year. Copyright © 2011 Pearson Education, Inc.

Two Boys and a Girl CN (3) 3. What is the probability that a family with three children has two boys and one girl? Assume boys and girls are equally likely. Copyright © 2011 Pearson Education, Inc.

Empirical and Subjective Probabilities An empirical probability is based on observations or experiments. It is the relative frequency of the event of interest. A subjective probability is an estimate based on experience or intuition. Copyright © 2011 Pearson Education, Inc.

500 Year Flood CN (4) Geological records indicate that a river has crested above a particular high flood level just four times in the past 2000 years. What is the empirical probability that the river will crest above this flood level next year? Copyright © 2011 Pearson Education, Inc.

Empirical Coin Tossing CN (5) If you are interested only in the number of heads in coin tosses, then the possible events when tossing two coins are 0 heads, 1 head, and 2 heads. Suppose you repeat a two-coin toss 100 times and your results are as follows. 0 heads occurs 22 times 1 head occurs 51 times 2 heads occurs 27 times Compare the empirical probabilities to the theoretical probabilities. 5. Do you have reason to suspect that the coins are unfair? Copyright © 2011 Pearson Education, Inc.

Three Types of Probabilities Theoretical probability The chance of rolling a 4 is 1 out of 6. Empirical probability Subjective probability Copyright © 2011 Pearson Education, Inc.

Three Types of Probabilities Theoretical probability She’s a 92% free throw shooter for the season. Empirical probability Subjective probability Copyright © 2011 Pearson Education, Inc.

Three Types of Probabilities Theoretical probability There’s about a 70% chance she will go out on a date with me. Empirical probability Subjective probability Copyright © 2011 Pearson Education, Inc.

Which Type of Probability? CN (6a-c) 6. Identify the method that resulted in the following statements: a. I’m 100% certain that you’ll be happy with this car. b. Based on data from the Department of Transportation, the chance of dying in an automobile accident during a one-year period is 1 in 7000a c. The chance of rolling a 7 with a twelve-sided die is 1/12. Copyright © 2011 Pearson Education, Inc.

Probability of an Event Not Occurring If the probability of an event A is P(A), then the probability that event A does not occur is 1 – P(A). Since the probability of a family of four children having two girls and two boys is 0.375, what is the probability of a family of four children not having two girls and two boys? P(not 2 girls) = 1 – 0.375 = 0.625 Copyright © 2011 Pearson Education, Inc.

Not Two Boys CN (7) 7. What is the probability that a randomly chosen family with three children does not have two boys and one girl? Assume boys and girls are equally likely. Copyright © 2011 Pearson Education, Inc.

Making a Probability Distribution A probability distribution represents the probabilities of all possible events. To make a probability distribution, do the following: Step 1: List all possible outcomes. Use a table or figure if it is helpful. Step 2: Identify outcomes that represent the same event and determine the probability of each event. Step 3: Make a table listing each event and probability. Copyright © 2011 Pearson Education, Inc.

A Probability Distribution All possible outcomes and a probability distribution for the sum when two dice are rolled are shown below. Possible outcomes Point out that the reason 7 is considered a lucky number in many gambling games is simply because there are more ways to roll a 7 than any other number. Copyright © 2011 Pearson Education, Inc.

Tossing Three Coins CN (8) 8. Make a table of the probability distribution for the number of heads that occur when three coins are tossed simultaneously. Copyright © 2011 Pearson Education, Inc.

Two Dice Distribution CN (9) Make a probability distribution for the sum of the dice when two dice are rolled. 9. What is the most probable sum? Copyright © 2011 Pearson Education, Inc.

Odds Odds are the ratio of the probability that a particular event will occur to the probability that it will not occur. The odds for an event A are . The odds against an event A are . Copyright © 2011 Pearson Education, Inc.

Two Coin Odds CN (10a-b) 10a. What are the odds for getting two heads in tossing two coins? b. What are the odds against it? Copyright © 2011 Pearson Education, Inc.

Horse Race Payoff CN (11) At a horse race, the odds on Blue Moon are given as 7 to 2. 11. If you bet $10 and Blue Moon wins, how much will you gain? Copyright © 2011 Pearson Education, Inc.

Quick Quiz CN (13) 13. Please answer the 10 quick quiz multiple choice questions on p.425. Copyright © 2011 Pearson Education, Inc.

Homework 7A Discussion Paragraph 6D P. 425-6: 1-12 1 web 70. Blood Groups 71. Accidents 1 world 72. Probability in the News 73. Probability in your Life 74. Gambling Odds Class Notes 1-13 Copyright © 2011 Pearson Education, Inc.