Transcript probability
Chapter 9 Introducing Probability BPS - 3rd Ed. Chapter 9 1 Idea of probability Probability is the science of chance behavior Chance behavior is unpredictable in the short run but has a predictable pattern in the long run BPS - 3rd Ed. Chapter 9 2 Randomness and probability A phenomenon is random if individual outcomes are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions. The probability of any outcome of a random phenomenon can be defined as the proportion of times the outcome would occur in a very long series of repetitions. BPS - 3rd Ed. Chapter 9 3 Thinking about probabilities The best way to understand randomness is to observe random behavior in a long run of independent trials Short runs give only rough estimates of probability BPS - 3rd Ed. Chapter 9 4 Empirical probabilities Coin flipping: eventually, the proportion approaches 0.5, the probability of a head BPS - 3rd Ed. Chapter 9 5 Exercise 9.5 (p. 227) Premise: Probability of 0 in the random number table is 0.1 a) What proportion of the first 50 digits is a 0? (ans: 3 of 50, or 0.06) b) Use the Probability Applet to simulate 40 at a time; set probability to 0.1. What is the result of 200 tosses? C:\Data\hs067\BPS3e\index.htm BPS - 3rd Ed. Chapter 9 6 Probability Models Skip this section (pp. 228 – 230) BPS - 3rd Ed. Chapter 9 7 Probability Rule 1 Any probability is a number between 0 and 1. A probability can be interpreted as the proportion of times that a certain event can be expected to occur. If the probability of an event is more than 1, then it will occur more than 100% of the time (Impossible!). BPS - 3rd Ed. Chapter 9 8 Probability Rule 2 All possible outcomes together must have probability 1. Because some outcome must occur on every trial, the sum of the probabilities for all possible outcomes must be exactly one. If the sum of all of the probabilities is less than one or greater than one, then the resulting probability model will be incoherent. BPS - 3rd Ed. Chapter 9 9 Probability Rule 3 The probability that an event does not occur is 1 minus the probability that the event does occur. If a person has a 0.75 chance of recovering, she must have a 1 – 0.75 = 0.250 chance of not recovering. If a person has a 0.95 chance of recovering, she must have a 1 – 0.95 = 0.05 chance of not recovering. BPS - 3rd Ed. Chapter 9 10 Probability Rule 4 If two events have no outcomes in common, they are said to be disjoint. The probability that one or the other of two disjoint events occurs is the sum of their individual probabilities. Age of woman at first child birth – under 20: 25% 24 or younger: 58% – 20-24: 33% – 25+: ? Rule 3 (or 2): 42% } BPS - 3rd Ed. Chapter 9 11 Probability Rules: Mathematical Notation BPS - 3rd Ed. Chapter 9 12 Assigning probabilities: finite Skip this section (pp. 232 – 235) BPS - 3rd Ed. Chapter 9 13 Assigning probabilities: intervals Recall: “areas under a density curve” (Chapter 3)! Illustration: random number generators give output (digits) spread uniformly across the interval from 0 to 1. Find the probability of getting a random number that is less than or equal to 0.5 OR greater than 0.8. P(X ≤ 0.5 or X > 0.8) = P(X ≤ 0.5) + P(X > 0.8) = 0.5 + 0.2 = 0.7 BPS - 3rd Ed. Chapter 9 14 Normal probability models The Normal curve the density curve that is most familiar to us – Normal random variable denoted X ~ N(µ, ) Technique for finding Normal probabilities covered in Chapter 3 – Convert observed values of the endpoints of the interval to Z scores – Find probabilities from Table A Example 9.9 in text (p. 237) BPS - 3rd Ed. Chapter 9 15 Random variable and Personal Probabilities Skip these sections (pp. 237 – 241) BPS - 3rd Ed. Chapter 9 16