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A (Brief) General Discussion of Probability: Some “Probability Rules” Some abstract math language too! (from various internet sources) Probability The Science of Random Behavior • Random behavior is unpredictable for an individual object, but it has a regular and predictable pattern on the average for a huge number of objects. • This is why we can use probability to gain useful results from random samples & randomized comparative experiments. • Random Individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions. 2 • Relative Frequency The proportion of occurrences of an outcome. It settles down to one value over the long run. That one value is then defined to be the probability of that outcome. • Relative Frequency Probabilities can be determined (checked) by observing a long series of independent trials (empirical data): – Do experiments with many samples – Do simulations, with computers, with random number tables, etc. 3 Example: Flipping a Fair Coin Probability Models • The sample space S of a random phenomenon is the set of all possible outcomes. • An event is an outcome or a set of outcomes (a subset of the sample space). • A probability model is a mathematical description of long-run regularity consisting of a sample space S and a method of assigning probabilities to events. Probability Model for Two Fair Dice Example of Random Phenomenon: Roll a pair of fair dice. The Sample Space is illustrated in the figure: The probabilities of each individual of the 36 outcomes are found by inspection. Each clearly occurs with a probability: p = (1/36) = 0.0278 Probability Rule #1: All Probabilities Must Be Numbers Between 0 & 1. • A probability can be interpreted as the proportion of times that a certain event can be expected to occur. • So, if the probability of an event is more than 1, then it will occur more than 100% of the time!! This is clearly illogical & impossible! Probability Rule #2: The sum of the probabilities of all possible outcomes must be 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 & illogical. Probability Rule #3: • 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. Example of Disjoint Probabilities: • If two events have no outcomes in common, they are disjoint. Probability that one or the other of 2 disjoint events occurs = sum of individual probabilities. • Consider data about the age distribution of women at first child birth. Obviously, the sample population is women have had children! So, it excludes women with no children! • A study of census data gives: Under 20: 25%. 20-24: 33%. 25+: ? • So, the probability that a woman in the sample population who is 24 or younger has had a first child is: = 25% + 33% = 58% Note: By Rule #3 (or Rule #2) This must be 42% Probability Rule #4: The probability that an event DOES NOT OCCUR 1 minus the probability that the event DOES OCCUR. • Example: As a jury member, you assess the probability that the defendant is guilty to be 0.8. So, you must also believe that the probability the defendant is not guilty is 0.2 in order to be consistent. • Similarly, if the probability that a flight will be on time is 0.7, then the probability it will be late is 0.3. Summary: Probability Rules in Mathematical Notation Consider again the Random Phenomenon of rolling a pair of fair dice. As we’ve already seen, The Sample Space is illustrated in the figure: The probabilities of each individual of the 36 outcomes are found by inspection. Each clearly occurs with a probability: p = (1/36) = 0.0278 All possible outcomes of rolling a pair of fair dice are shown in the figure. Calculate the probability P(5) of rolling a 5. All possible outcomes of rolling a pair of fair dice are shown in the figure. Calculate the probability P(5) of rolling a 5. P(5) P( ) + P( ) + P( ) + P( ) = All possible outcomes of rolling a pair of fair dice are shown in the figure. Calculate the probability P(5) of rolling a 5. P(5) P( ) + P( ) + P( ) + P( ) = (1/36) + (1/36) + (1/36) + (1/36) = (4/36) = = 0.111