1. Lesson 5 - 1 Probability Rules

2. 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

3. 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

4. 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

5. 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.

6. 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, E i ’s, must equal 1 ∑ P(E i ) = 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

7. 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

8. 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

9. Example 1 Using a six-sided dice, answer the following: a) P(rolling a six) b) P(rolling an even number) b) P(rolling 1 or 2) d) P(rolling an odd number) 1/6 3/6 or 1/2 2/6 or 1/3 3/6 or 1/2

10. Example 2 Identify the problems with each of the following a) P(A) =.35, P(B) =.40, and P(C) =.35 b) P(E) =.20, P(F) =.50, P(G) =.25 c)P(A) = 1.2, P(B) =.20, and P(C) =.15 d)P(A) =.25, P(B) = -.20, and P(C) =.95 ∑P > 1 ∑P < 1 P() > 1 P() < 0

11. 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