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Chapter 6: Probability 6.1 Introduction The probability of an event is a number between 0 and 1 that expresses the long-run likelihood that the event will occur. An event having probability 0.1 is rather unlikely to occur. An event with probability 0.9 is very likely to occur. An event with probability 0.5 is just as likely to occur as not. Example: A clinic tests for active pulmonary tuberculosis. If a person has tuberculosis, the probability of a positive test result is 0.98. If a person does not have tuberculosis, the probability of a negative test result is 0.99. The incidence of tuberculosis in a certain city is 2 cases per 10,000 population. What is the probability that an individual who tests positive actually has pulmonary tuberculosis? 6.2 Experiments, Outcomes, and Events An experiment is an activity with an observable outcome. Each repetition of the experiment is called a trial. In each trial we observe the outcome of the experiment. Experiment 1: Flip a coin Trial: One coin flip Outcome: Heads Experiment 2: Allow a conditioned rat to run a maze containing three possible paths Trial: One run Outcome: Path 1 Experiment 3: Tabulate the amount of rainfall in Ceres, CA in a year Trial: One year Outcome: 11.23 in. The sample space of an experiment is the set of all possible outcomes of the experiment. Example: An experiment consists of throwing two dice, one red and one green, and observing the numbers on the uppermost face on each. What is the sample space S of this experiment? An event E is a subset of the sample space. Example: For the experiment of rolling two dice, describe the following events: E1 = {The sum of the numbers is greater than 9}; E2 = {The sum of the numbers is 7 or 11}. Let S be the sample space of an experiment. The event corresponding to the empty set, is called the impossible event, since it can never occur. The event corresponding to the sample space itself, S, is called the certain event because the outcome must be in S. Let E and F be two events of the sample space S. The event where either E or F or both occurs is designated by E ∪ F. The event where both E and F occurs is designated by E ∩ F. The event where E does not occur is designated by E '. For the experiment of rolling two dice, let E1 = “The sum of the numbers is greater than 9” and E3 = “The numbers on the two dice are equal”. Determine the sets E1 ∪ E3, E1 ∩ E3, and (E1 ∪ E3)'. 6.3 Assignment of Probabilities Suppose you took a coin and tossed it 200 times. Number Relative frequency Heads 68 68/200 = 34% Tails 132 132/200 = 66% Total 200 1 or 100% The experimental probability that heads occurs is 34% and that tails occurs is 66%. Probability Distribution for the roll of a die Outcome 1 2 3 Probability 1/6 1/6 1/6 Outcome 4 5 6 Probability 1/6 1/6 1/6 Traffic engineers measure the volume of traffic on a major highway during the rush hour. Generate a probability distribution using the data generated over 300 consecutive weekdays. Assign a probability distribution to this experiment. Let an experiment have outcomes s1, s2, … , sN with probabilities p1, p2, … , pN. Then the numbers p1, p2, … , pN must satisfy: Fundamental Property 1 Each of the numbers p1, p2, … , pN is between 0 and 1; Fundamental Property 2 p1 + p2 + … + pN = 1. Addition Principle Suppose that an event E consists of the finite number of outcomes s, t, u, … ,z. That is E = {s, t, u, … ,z }. Then Pr(E) = Pr(s) + Pr(t) + Pr(u) + … + Pr(z), Inclusion-Exclusion Principle Let E and F be any events. Then Pr( E F ) Pr( E ) Pr( F ) Pr( E F ). If E and F are mutually exclusive, then Pr( E F ) Pr( E ) Pr( F ). Converting between odds and probability If the odds in favor of an event E are a to b, then a b Pr( E ) and Pr( E ) . ab ab On average, for every a + b trials, E will occur a times and E will not occur b times. 6.4 Calculating Probabilities of an Event Let S be a sample space consisting of N equally likely outcomes. Let E be any event. Then Pr(E ) number of outcomes in E . N Complement Rule Let E be any event, E ' its complement. Then Pr(E) = 1 - Pr(E '). 6.5 Conditional Probability and Independence Let E and F be events is a sample space S. The conditional probability, Pr( E | F ) is the probability of event E occurring given the condition that event F has occurred. In calculating this probability, the sample space is restricted to F. Pr( E F ) Pr( E | F ) Pr( F ) provided that Pr(F) ≠ 0. Product Rule If Pr(F) ≠ 0, Pr(E ∩ F) = Pr(F) Pr(E | F). The product rule can be extended to three events. Pr(E1 ∩ E2 ∩ E3) = Pr(E1) Pr(E2 | E1) Pr(E3| E1 ∩ E2) Let E and F be events. We say that E and F are independent provided that Pr(E ∩ F) = Pr(E) Pr(F). Equivalently, they are independent provided that Pr(E | F) = Pr(E) and Pr(F | E) = Pr(F). A set of events is said to be independent if, for each collection of events chosen from them, say E1, E2, …, En, we have Pr(E1 ∩ E2 ∩ … ∩ En) = Pr(E1) Pr(E2) … Pr(En).