CHAPTER 7 Random Phenomena cannot predict the individual outcomes but can hope to understand characteristics of their long run behavior A Phenomena consists of trials Each trial has an outcome Outcomes combine to make events Sample Space Collection of all possible outcomes The Probability of an event is its long run relative frequency A relative frequency is a fraction which means we can write it as a fraction decimal or a percent Independent the outcome of one trial doesn t influence or change the outcome of another Law of Largo Numbers LLN If events are independent then as the number of trials increases the long run frequency of an event occurring gets closer and closer to a single value Independent random events make each individual result impossible to predict Empirical Theory a relative frequency of an event s occurrence as the probability of an event The probability of an event is the number of outcomes in the event divided by the total number of possible outcomes Personal Probability Based on experience they are not based either on long run relative frequencies or on equal likely events 1 If the probability of an event occurring is 0 the event CANNOT occur likewise if the probability is 1 the event ALWAYS occurs 2 Something always occurs so the probability of something 3 The set of outcomes that are not in the event is called the happening is 1 compliment of the event o The probability of an event occurring is 1 minus the probability that it doesn t occur 4 Multiplication Rule says that to find the probability that 2 independent events occur we multiply the probabilities o 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 o P A and B P A x P B 5 Only having the option to choose one event of two makes the outcomes disjoint PROBABILITY RULES o The addition rule allows us to add the probabilities of disjoint events to get the probability that either event occurs o P A or B P A P B 6 General Addition Rule which does not require disjoint events o P A or B P A P B P A and B Marginal Probability uses a marginal frequency From either the Total Row or Total Column to compute the probability Joint probabilities Are the probabilities of two events occurring together Conditional Probability The probability of B given A or P BlA o P BlA P A and B P A General Multiplication Rule Does not require the events to be independent P A and B P A x P BlA Events A and B are independent when o P BlA P B Whenever at least one of the joint probabilities in the table is not equal to the product of the marginal probabilities we say that the variables are not independent DISJOINT events CANNOT be INDEPENDENT
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