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Probability Basics

This activity involves rolling two dice to determine the sums and explore probability concepts such as sample space, multiplication principle, complements, and disjoint events.

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Probability Basics

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  1. Probability Basics Section 6.2.1

  2. Starter 6.2.1 • Roll two dice and record the sum shown. • Repeat until you have done 20 rolls. • Write a list of all the possible outcomes (sums) and how many of each you got. • Report those totals to me when I call for them. • I will record class totals and show the distribution of outcomes.

  3. Objectives • Write the sample space for a set of events. • Count the outcomes in a sample space by using the multiplication principle. • Describe what is meant by the complement of an event. • Describe what it means to say that two events are disjoint. • Use the addition rule for disjoint events to answer “or” probability questions.

  4. The Sample Space • The sample space of an event is a listing of all possible outcomes. • List all the sums possible in the starter. S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} • Suppose the question had asked you to do 20 rolls of two dice and note the individual dice face-up spots as ordered pairs • Example: (red 3, green 6) • Write the sample space for rolling two dice • In other words, write an organized list or table showing all the possible ordered pairs

  5. The Sample Space of Two Dice • Notice that each of the 36 outcomes is equally likely • The sums are not equally likely because each sum has different numbers of outcomes above • Compare theoretical probability of each sum with the empirical probabilities we found in the starter

  6. The Multiplication Principle • If task A can be done in x ways, and task B can be done in y ways, then the number of ways to do both A and B is • There were 6 ways to roll the red die and 6 ways to roll the white die, so there are 36 ways to roll both dice. • In yesterday’s starter, there were 2 ways to toss the first coin and 2 ways to toss the second coin and 2 ways to toss the third coin, so there are 8 ways to toss the three coins.

  7. The Rules of Probability • Rule 1: For any event A: • Rule 2: If S is the sample space: P(S) = 1 • Rule 3: The complement of event A is the event that A does not occur • The Complementary Probability Rule is: P(Ac) = 1 – P(A) • Rule 4: Two events A and B are disjoint if they are mutually exclusive (if A happens, B cannot and vice-versa). • The probability of two disjoint events is: P(A or B) = P(A) + P(B) • This is called the “disjoint or” rule

  8. Disjoint Examples • Roll two dice. I will pay you $1 if you get a sum of 7 or 8. • What is event A? • What is event B? • Are A and B disjoint? • What is P(A or B)? • Choose a student from this class at random. I will pay you $1 if the student is a senior or a boy. • What is event A? • What is event B? • Are A and B disjoint?

  9. Objectives • Write the sample space for a set of events. • Count the outcomes in a sample space by using the multiplication principle. • Describe what is meant by the complement of an event. • Describe what it means to say that two events are disjoint. • Use the addition rule for disjoint events to answer “or” probability questions.

  10. Homework • Read pages 318 – 329 • Do problems 9, 12, 15, 18, 20-23

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