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O =CSC-141 Lecture 17 ProbabilityThe Sample SpaceA sample space has two parts: A list of outcomes. An assignment of probabilities to these outcomes. In the case of a fair die (like the ones they use in casinos): The outcomes are 1, 2, 3, 4, 5, and 6. The probability of each is 1/6, and the sum of the probabilities for all six is 1.TF?zF?zMore on Sample SpaceMore specifically, the sample space consists of a set and a function. The set is the list of all possible outcomes of some experiment. The function assigns a numerical score to each outcome. This numerical score  called the probability of the outcome  is simply a real number between 0 and 1 (inclusive). We also require the sum of the probabilities of all the outcomes to be exactly 1.ZNomenclature and the Die We use the letter S for the set of outcomes, and the letter P for the function that assigns to each s S the probability of that outcome, P(s).)'" Spinner  Pair of DiceDie 1 can land in any of six equally likely ways. Die 2 can also land in any of six equally likely ways, independent of die 1. We can express the outcome as an ordered pair, where there are 36 possible outcomes. Each is equally likely.,ZioEventsAn event is a collection of outcomes. An event A is a subset of S (that is, A S). The probability of an event is:lt/ #Rolling Ten DiceTen dice are rolled. What is the probability that none of the dice show the number 1? We begin by constructing a sample space and let S denote the set of all possible outcomes. There are 610 such lists. Event A has 510 elements.l5    Four ChildrenRA couple has four children. Which is more likely: They have two boys and two girls. They have three of one gender and one of the other. Set S consists of 16 possible outcomes, including four boys and four girls (2x2x2x2). Event A is 2 of each: {ggbb,gbgb,gbbg,bbgg,bgbg,bggb} Event B is 3 of one: {gggb,ggbg,gbgg,bggg,bbbg,bbgb,bgbb,gbbb}B2ZVZZ2V Combining Events*Union of Events. If A and B are events, so is A B. We can think of this as the event that A or B occurs. Intersection of Events. Likewise, A B is an event that represents when both A and B occur. Difference of Events. The set A  B is the event that A occurs, but B does not. Complement of an Event. Since S is the  universe of all outcomes, then ~A is the same as S  A. ~A is when A does not occur.Z)+'& )*  Probability of Combined Events The Birthday Problem, Part 1Four people are chosen at random. What is the probability that two of them have the same birthday? To simplify the problem, we make two assumptions: We ignore February 29. We assume that each day of the year is equally likely as a birthday..Z\Z\ The Birthday Problem, Part 2With the same simplifying assumptions, what is the probability of two people having the same birthday from a randomly selected set of 23 people? Conditional ProbabilityLet A represent the event that a student misses the school bus. Let B represent the event that the student s alarm clock malfunctions. Both events have a low probability, however, what is the probability of the student missing the bus given the fact that the alarm clock malfunctioned? We denote this as P(A|B)  the probability that event A occurs given that event B occurs.xZ? Independent EventsLet A and B be events in a sample space. Suppose that P(A) and P(B) are both non-zero. The statements on the right are equivalent:~+A!What are the Odds?Odds are simply another way of looking at probabilities. Odds are a ratio of probabilities. The probability of a fair coin being a head is 0.5, and the probability of it being a tail is also 0.5. Therefore the odds are 1:1. The probability of a fair die being a 1 is 1/6, while the probability of it being something else is 5/6: The odds of rolling a 1 are 1:5 in favor, or 5:1 against. 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