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Warm Up Express the indicated degree of likelihood as a probability value: “There is a 40% chance of rain tomorrow.” A bag contains 6 red marbles, 3 blue marbles, and 7 green marbles. If a marble is randomly selected from the bag, what is the probability that it is blue? Assume that one student in a class of 27 students is randomly selected to win a prize. Would it be “unusual” for you to win? (Assume “unusual” is a probability less than or equal to 0.05)

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**Chapter 4 Probability 4-1 Review and Preview**

4-2 Basic Concepts of Probability 4-3 Addition Rule 4-4 Multiplication Rule: Basics 4-5 Multiplication Rule: Complements and Conditional Probability 4-6 Counting 4-7 Probabilities Through Simulations 4-8 Bayes’ Theorem

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Objective Students will use the addition rule for finding probabilities of the form P(A or B).

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P(A or B) P(A or B) is the probability that either event A occurs or event B occurs (or they both occur). The key word in this section is “or.” In mathematics we usually use the inclusive or, which means A ,B or both.

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**Compound Event Compound Event**

any event combining 2 or more simple events Notation A or B is a compound event because it combines the two simple events A and B. We are moving from studying probabilities of single events to probabilities of compound events.

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**Algorithm for finding P(A or B)**

1. Find the total number of ways A can occur. 2. Find the total number of ways B can occur. 3. Find the total number of ways A and B occur together. (the overlap P(A and B)) 4. Find the size of the sample space. 4. Add the probability of A to the probability of B and subtract the overlap. Subtracting the overlap prevents double counting)

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Recall

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Specifically Formal Addition Rule P(A or B) = P(A) + P(B) – P(A and B)

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**Pre – Employment Drug Screening Results**

Positive test result Negative test result Subject uses drugs 44 6 Subject does not use drugs 90 860 Find the probability P(positive test result or subject uses drugs) There are 134 positive test results out of 1000 results total so P(positive test result) = 134/1000 = 67/500 There are 50 drug user results out of 1000 results total so P(positive test result) = 50/1000 = 25/500 There are 44 results who use drugs and tested positive and out of 1000 total results total so P(positive test result and drug user) = 44/1000 = 22/500

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**So P(positive test result or subject uses drugs)**

= 67/500 +25/500 -22/500 = 70/500 = 7/50

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Caution When finding P(A or B), be sure to add in such a way that every outcome is counted only once. Watch out for double counting.

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**Disjoint (Mutually Exclusive)**

Two events A and B are disjoint (or mutually exclusive) if they cannot occur at the same time. Disjoint events do not overlap.

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**Decide if the events are disjoint or not.**

A =The event of randomly selecting someone taking a statistics course. B =The event of randomly selecting someone who is a female. A and B are not disjoint A =The event of randomly selecting someone who is a Registered Democrat. B =The event of randomly selecting someone who is a registered Republican. A and B are disjoint

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**Disjoint (A visual perspective)**

Venn Diagram for Events That Are Not Disjoint Venn Diagram for Disjoint Events

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Alternate notation

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**P(A or B) (A visual perspective)**

Venn Diagram for Events That Are Not Disjoint Venn Diagram for Disjoint Events

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**Complementary Events Recall A and are complements.**

Since it is impossible for both to occur together at the same time, they must be disjoint.

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**Complements (A visual perspective)**

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Example Based on data from a Harris Interactive Poll, the probability of randomly selecting someone who believes in the devil is 0.6. P(believes in the devil) = 0.6. Use this to find P(person does not believe in the devil)

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