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Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. SECTION 4.1 BASIC IDEAS Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Objectives 1. 2. 3. Construct sample spaces Compute and interpret probabilities Approximate probabilities using the Empirical Method Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Objective 1 Construct sample spaces Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Probability Experiment A probability experiment is one in which we do not know what any individual outcome will be, but we do know how a long series of repetitions will come out. For example, if we toss a fair coin, we do not know what the outcome of a single toss will be, but we do know what the outcome of a long series of tosses will be – about half “heads” and half “tails”. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Probability The probability of an event is the proportion of times that the event occurs in the long run. So, for a “fair” coin, that is, one that is equally likely to come up heads as tails, the probability of heads is 1/2 and the probability of tails is 1/2. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Law of Large Numbers The law of large numbers says that as a probability experiment is repeated again and again, the proportion of times that a given event occurs will approach its probability. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Sample Space The collection of all the possible outcomes of a probability experiment is called a sample space. Example: Suppose that a coin is tossed. The sample space consists of: {Heads, Tails} Suppose that a standard die is rolled. The sample space consists of: {1, 2, 3, 4, 5, 6} Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Event We are often concerned with occurrences that consist of several outcomes. For example, when rolling a die, we might be concerned with the possibility of rolling an odd number. A collection of outcomes of a sample space is called an event. Example: A probability experiment consists of rolling a die. The sample space is {1, 2, 3, 4, 5, 6}. The event of rolling an odd number = {1, 3, 5}. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Probability Model Once we have a sample space for an experiment, we need to specify the probability of each event. This is done with a probability model. We use the letter “P” to denote probabilities. For example, if we toss a coin, we denote the probability that the coin lands heads by “P(Heads).” Notation: If A denotes an event, the probability of event A is denoted by P(A). Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Probabilities With Equally Likely Outcomes If a sample space has n equally likely outcomes, and an event A has k outcomes, then Number of outcomes in A k P(A) Number of outcomes in the sample space n Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Objective 2 Compute and interpret probabilities Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example A fair die is rolled. Find the probability that an odd number comes up. Solution: The sample space has six equally likely outcomes: {1, 2, 3, 4, 5, 6} The event of an odd number has three outcomes: {1, 3, 5} The probability is: 3 1 P(Odd Number) 6 2 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example A family has three children. Denoting a boy by B and a girl by G, we can denote the genders of these children from oldest to youngest. For example, GBG means the oldest child is a girl, the middle child is a boy, and the youngest child is a girl. There are eight possible outcomes: BBB, BBG, BGB, BGG, GBB, GBG, GGB, and GGG. Assume these outcomes are equally likely. What is the probability that all three children are the same gender? Solution: Of the eight equally likely outcomes, the two outcomes BBB and GGG correspond to having all children of the same gender. Therefore 2 1 P(All three have same gender) 8 4 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Probability Rules The probability of an event is always between 0 and 1. That is, 0 ≤ P(A) ≤ 1. If A cannot occur, then P(A) = 0. If A is certain to occur, then P(A) = 1. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Sampling From a Population Sampling an individual from a population is a probability experiment. The population is the sample space and members of the population are equally likely outcomes. Example: There are 10,000 families in a certain town categorized as follows: Own a house Own a condo Rent a house Rent an apartment 4753 1478 912 2857 A pollster samples a single family from this population. What is the probability that the sampled family rents? Solution: The number of families who rent is 912 + 2857 = 3769. Therefore, the probability that the sampled family rents is 3769/10,000 = 0.3769. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Unusual Events An unusual event is one that is not likely to happen. In other words, an event whose probability is small. A rule of thumb is that any event whose probability is less than 0.05 is considered to be unusual. Example: In a college of 5000 students, 150 are math majors. A student is selected at random and turns out to be a math major. Is this an unusual event? Solution: The event of choosing a math major consists of 150 students out of a total of 5000 students. The probability of choosing a math major is 150/5000 = 0.03. Since 0.03 < 0.05, this would be considered an unusual event. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Objective 3 Approximate probabilities using the Empirical Method Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Approximating Probabilities with the Empirical Method The law of large numbers says that if we repeat a probability experiment a large number of times, then the proportion of times that a particular outcome occurs is likely to be close to the true probability of the outcome. The Empirical Method consists of repeating an experiment a large number of times, and using the proportion of times an outcome occurs to approximate the probability of the outcome. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example The Centers for Disease Control reports that in the year 2002 there were 2,057,979 boys and 1,963,747 girls born in the U.S. Approximate the probability that a newborn baby is a boy. Solution: The number of times that the experiment has been repeated is: 2,057,979 boys + 1,963,747 girls = 4,021,726 births The proportion of births that are boys is: 2,057,979/4,021,726 = 0.5117 Therefore, the probability that a newborn baby is a boy is approximated by 0.5117. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Do You Know… • • • • How to construct a sample space? How to compute probabilities of equally likely events? The rules of probability? How to compute probabilities using the Empirical Method? Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.