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Understanding Probability and Complement Events in Statistics

This lesson focuses on finding the probability of the complement of events in probability theory. Students will learn that the complement of event E (denoted E′) consists of outcomes not included in E, with the key relationship P(E) + P(E′) = 1. Through various examples, such as rolling dice, selecting cards, and using tree diagrams, learners will explore applications of probability calculations. The Fundamental Counting Principle will also be reviewed to understand how to calculate total outcomes in sample spaces effectively.

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Understanding Probability and Complement Events in Statistics

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  1. Introductory Statistics Lesson 3.1 D Objective: SSBAT find the probability of the complement of events and applications of probability. Standards: M11.E.3.1.1

  2. Complement of Event E • The set of all outcomes in a sample space that are NOT included in event E • The complement of event E is denoted by E′ • E′ is read as “E prime” • P(E) + P(E′) =1

  3. Example: • Roll a die and let E be the event of rolling a 1 or 2. • E′ would then be rolling a 3, 4, 5, 6 • E = {1, 2} • E′ = {3, 4, 5, 6}

  4. Examples. Use the spinner to the right. Find the probability of not rolling a 5. P(not 5) = = P(not 7 or 8) =

  5. Use a standard deck of cards. Find the Probability of not picking a Heart P(Not Heart) = = or 0.75

  6. You put all the letters of the alphabet in a hat. You randomly pick one letter from the hat. What is the probability that you do not pick a vowel? (there are 5 vowels in the alphabet)

  7. Sometimes you will have to use a Tree Diagram or the Fundamental Counting Principle to find the total number in the sample space first before finding the probability.

  8. Review: Fundamental Counting Principle • How many ways can a committee of 5 people be chosen from a group of 30 people? • ____ ____ ____ ____ ____ 30 · 29 · 28 · 27 · 26 = 17,100,720 17,100,720 different ways

  9. Review: Tree Diagram • Find the sample space for choosing an outfit from the following. • Shirt: Sweater, Blouse, T-Shirt • Pants: Jeans or Khakis

  10. Example with a Tree Diagram: • Samantha tosses 3 dimes into the air. What is the Probability of Exactly 2 Heads. • Make a tree diagram to show the possible outcomes • Possible Outcomes: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} • P(2 Heads) =

  11. A customer has the following options for purchasing a new car. • Manufacturer: Ford, Chevrolet, Dodge • Doors: 2 door or 4 door • Colors: Red, Black, Silver • What is the probability that the next car sold is a 4 door? • Find the possible outcomes using tree diagram. b) What’s the probability that the next car sold is a Red Chevy?

  12. Examples with the Fundamental Counting Principle The daily number in the PA lottery consists of 3 numbers. Each number can be from 0 to 9 and the numbers may repeat. If you randomly choose a 3 digit number to play, what is the probability you will pick the winning number?  Find how many possible outcomes there are  10 · 10 · 10 = 1,000  P(winning) =

  13. You roll 2 dice. • What is the probability of getting the same number on each die. •  Make a tree diagram showing the possible outcomes • P(Same #) = • P(Same #) = or 0.167

  14. Homework Worksheet 3.1 D

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