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Lesson 5 - 1 Probability Rules Objectives • Understand the rules of probabilities • Compute and interpret probabilities using the empirical method • Compute and interpret probabilities using the classical method • Use simulation to obtain data based on probabilities • Understand subjective probabilities Vocabulary • Probability – measure of the likelihood of a random phenomenon or chance behavior • Outcome – a specific value of an event • Experiment – any process with uncertain results that can be repeated • Sample space – collection of all possible outcomes • Event – is any collection of outcomes for a probability experiment Vocabulary • Probability model – lists the possible outcomes of a probability experiment and each outcome’s probability • Impossible – probability of the occurrence is equal to 0 • Certainty – probability of the occurrence is equal to 1 • Unusual Event – an event that has a low probability of occurring • Tree Diagram – a list of all possible outcomes • Subjective Probability – probability is obtained on the basis of personal judgment The Law of Large Numbers As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed get closer to the probability of the outcome. Rules of Probability The probability of any event E, P(E), must between 0 and 1 0 ≤ P(E) ≤ 1 The sum of all probabilities of all outcomes, Ei’s, must equal 1 ∑ P(Ei) = 1 A more sophisticated concept: An unusual event is one that has a low probability of occurring This is not precise … how low is “low? Typically, probabilities of 5% or less are considered low … events with probabilities of 5% or lower are considered unusual Empirical Approach The probability of an event is approximately the number of time event E is observed divided by the number of repetitions of the experiment Frequency of E P(E) ≈ relative frequency of E = ---------------------------------Total Number of Trials Classical Method If an experiment has n equally likely outcomes and if the number of ways that an event E can occur is m, then the probability of E, P(E), is Number of ways that E can occur m P(E) = -------------------------------------------- = -------Number of possible outcomes n Example 1 Using a six-sided dice, answer the following: a) P(rolling a six) 1/6 b) P(rolling an even number) 3/6 or 1/2 b) P(rolling 1 or 2) 2/6 or 1/3 d) P(rolling an odd number) 3/6 or 1/2 Example 2 Identify the problems with each of the following a) P(A) = .35, P(B) = .40, and P(C) = .35 ∑P > 1 b) P(E) = .20, P(F) = .50, P(G) = .25 ∑P < 1 c) P(A) = 1.2, P(B) = .20, and P(C) = .15 P() > 1 d) P(A) = .25, P(B) = -.20, and P(C) = .95 P() < 0 Summary and Homework • Summary – Probabilities describe the chances of events occurring … events consisting of outcomes in a sample space – Probabilities must obey certain rules such as always being greater than or equal to 0 – There are various ways to compute probabilities, including empirically, using classical methods, and by simulations • Homework – pg 261-265; 9, 11, 12, 15, 18, 25, 26, 32, 34