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From Randomness to Probability 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. Ex: What light (red, yellow, green) will you encounter when you drive to an intersection? Toss a coin. ◦ It’s equally likely to get heads or tails ◦ Sample space? Roll a die. ◦ It’s equally likely to get any one of six outcomes ◦ Sample space ? The events are not always equally likely. ◦ A skilled basketball player has a better than 50-50 chance of making a free throw. ◦ What’s the chance that the light will be green as you drive to an intersection? Toss two coins: Sample space ? Toss three coins: Sample space ? The probability of an event A is # of outcomes in A P (A) = # of possible outcomes When thinking about what happens with combinations of outcomes, things are simplified if the individual trials are independent. ◦ This means that the outcome of one trial doesn’t influence or change the outcome of another. For example, coin flips are independent. How about drawing cards from a deck of 52 cards? Empirical Probability We used a computer to perform two simulations of tossing a balanced coin 100 times. Note that in a large number of tosses, the coin will land with heads facing up about half the time. The Law of Large Numbers (LLN) says that the long-run relative frequency of repeated independent events gets closer and closer to a single value. This single value is called the probability of the event. Because this definition is based on repeatedly observing the event’s outcome, this definition of probability is often called empirical probability. In everyday speech, when we express a degree of uncertainty without basing it on long-run relative frequencies or mathematical models, we are stating subjective or personal probabilities. Personal probabilities don’t display the kind of consistency that we will need probabilities to have, so we’ll stick with formally defined probabilities. The LLN says nothing about short-run behavior. Relative frequencies even out only in the long run, and this long run is really long (infinitely long, in fact). The so called Law of Averages (that an outcome of a random event that hasn’t occurred in many trials is “due” to occur) doesn’t exist at all. Example When two balanced dice are rolled, what’s the probability of rolling a sum of 8? The sum of the dice can be 8 in five ways. The probability the sum is 8 is 5/36. The most common kind of picture to make is called a Venn diagram. Example: 120 students are surveyed about what subjects they like at school. 40 students like biology, 30 students like chemistry. 25 like both biology and chemistry. How many students like neither of the subjects? 1. Probability Rule: ◦ ◦ A probability is a number between 0 and 1. For any event A, 0 ≤ P (A) ≤ 1. 2. Probability Assignment Rule: ◦ The probability of the set of all possible outcomes of a trial must be 1. P (S) = 1 (S is sample space.) 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) 4. Addition Rule: ◦ Events that have no outcomes in common (and, thus, cannot occur together) are called disjoint (or mutually exclusive). ◦ 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. ◦ 5. Multiplication Rule: ◦ ◦ ◦ 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), Two independent events A and B are not disjoint, provided the two events have probabilities greater than zero Examples 1) According to tables provided by the U.S. National Center for Health Statistics in Vital Statistics of the United States, a person aged 20 has about an 80% chance of being alive at age 65. Suppose that three people aged 20 are selected at random. What’s the probability that all three people will be alive at age 65? 2) For a sales promotion, the manufacturer places winning symbols under the caps of 5% of all Pepsi bottles. You buy a six-pack. What is the probability that you win something? 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) Examples 1) A quiz consists of 5 multiple-choice questions, each with 4 possible answer. If you have no clue and randomly guess answers, what’s the probability that you get a) a perfect score? b) exactly one correct answer? c) a 0 score? d) at least one correct answer? Page 379 – 383 Problem # 1, 9, 11, 13, 15, 17, 19, 21, 25, 27, 31, 33, 35, 39, 41. Birthday problem