Probability Models Definitions probability model – describes all possible outcomes and says how to assign probabilities to any collection of outcomes sample space – collection of unique outcomes of a random circumstance – we usually denote this by S event – a collection of outcomes Suppose you are asked to toss a coin. What is the sample space? You toss the coin and it lands on Tails. What is the outcome? If you toss the coin three times and it lands THH then THH is the event. Probability Rules Rule 1. Any probability is a number between 0 and 1. So if we observe an event A then we know 0 ≤ P(A) ≤ 1 Rule 2. All possible (disjoint or non-overlapping) outcomes together must have probability 1. • An outcome must occur on every trial. • The sum of the probabilities for all possible outcomes must be exactly 1. Consider the following assignment of probabilities Marital Status of a Random Sample of Women Ages 25 to 29 Marital Status Probability Never married 0.386 Married 0.555 Widowed 0.004 Divorced 0.055 • Each of the probabilities is a number between 0 and 1. • The probabilities total to 1. 0.386 + 0.555 + 0.004 + 0.055 = 1 • Any assignment of probabilities to all individual outcomes that satisfies Rules 1 and 2 is legitimate. Rule 3. The probability that an event does not occur is 1 minus the probability that the event does occur. This is known as the complement rule. • Suppose that P(A) = .70Chapter 18 Page 2 • Using this rule we can determine P(“not” A). The event “not A” is known as the complement of A which can be written as AC P(“not” A) = Clicker Questions about Complements and their probabilities Suppose the probability of a horse winning a race is 0.85. What is the complement of the horse winning? A. Horse winning B. Horse not winning C. 0.85 D. 0.15 What is the probability of the horse not winning? A. Horse winning B. Horse not winning C. 0.85 D. 0.15 Rule 4. Union Rule: If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. If this is true then the events are said to be disjoint. Suppose events A and B are disjoint and you know that P(A) = .40 and P(B) = .35. What is the P(A or B)? A Venn diagram can help us picture events and sample spaces.Chapter 18 Page 3 Example illustrating more general form of Rule 4 The probability a student is in honors math is 0.25, the probability a student is in honors science is 0.3, and the probability a student is in both is 0.2. What is the probability a student is in at least one honors class? This is called a union and can be pictured on a Venn diagram Probability of honors math or science (or both) = P(honors math or honors science) The probability it will rain Wednesday AM is 30%. The probability it will rain Wednesday PM is 30%. The probability it will rain both Wednesday AM and Wednesday PM is 10%. What is the probability it will rain on Wednesday? A) 20% D) 50% B) 30% E) 60% C) 40% Rule 5. Multiplication Rule: If two events, A and B, are independent then P(A and B) = P(A)P(B) Independence means that the occurrence of event A does not affect the occurrence of event B What is the probability that the first roll of a die is even, and the second roll is odd? A – 1/36 B – 1/12 C – 1/4 D – 1/2Chapter 18 Page 4 Sampling Distribution of sampling distribution – tells what values a statistic takes in repeated samples from the same population • We can’t predict the outcome of one sample, but the outcomes of many samples have a regular pattern. • The distribution of the statistic tells us what values it can take and how often it takes those values Take an SRS of 1523 adults. Ask each if they bought a lottery ticket in the last 12 months. The sample proportion who say “yes” is Suppose the true proportion, p = 0.60. If we were to take 1000 SRSs and calculate 1000 ’s the distribution would look like the picture to the right…. Example An opinion poll asks an SRS of 501 teens, “Generally speaking, do you approve or disapprove of legal gambling or betting?” Suppose that in fact exactly 50% of all teens would approve if asked (so p = 0.5). Suppose the sample proportion, , who approve is normally distributed with mean 0.5 and standard deviation 0.022. What is the probability that a poll will indicate less than 47.8% of teens
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