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Understanding Disjoint Events and Probability

Learn about disjoint events and how to calculate the probability of either event occurring, as well as the probability of one event following another in independent trials.

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Understanding Disjoint Events and Probability

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  1. California Standards SDAP3.4 Understand that the probability of either of two disjoint events occurring is the sum of the two individual probabilities and that the probability of one event following another, in independent trials, is the product of the two probabilities. Also covered:SDAP3.1

  2. Vocabulary disjoint events

  3. On a game show, the letters in the word Hollywood are printed on cards and shuffled. A contestant will win a trip to Hollywood if the first card she chooses is printed with an O or an L. Choosing an O or an L on the first card is an example of a set of disjoint events. Disjoint eventscannot occur in the same trial of an experiment.

  4. Additional Example 1: Identifying Disjoint Events Determine whether each set of events is disjoint. Explain. A. choosing a dog or a poodle from the animals at an animal shelter The event is not disjoint. A poodle is a type of dog, so it is possible to choose an animal that is both a dog and a poodle. B. choosing a fish or a snake from the animals at a pet store The event is disjoint. Fish and snakes are different types of animals, so you cannot choose an animal that is both a fish and a snake.

  5. Check It Out! Example 1 Determine whether each set of events is disjoint. Explain. A. choosing a bowl of soup or a bowl of chicken noodle soup from the cafeteria The event is not disjoint. Chicken noodle is a type of soup, so it is possible to choose a bowl of chicken noodle soup and soup. B. choosing a bowl of chicken noodle soup or broccoli cheese soup The event is disjoint. Chicken noodle and broccoli cheese are different types of soups, so you cannot choose a soup that is both chicken noodle and broccoli cheese.

  6. Reading Math Disjoint events are sometimes called mutually exclusive events. Probability of Two Disjoint Events P(A or B) = P(A) + P(B) Probability of either event Probability of one event Probability of other event

  7. 1 11 P(E) = Additional Example 2: Finding the Probability of Disjoint Events Find the probability of each set of disjoint events. A. choosing an A or an E from the letters in the word mathematics 2 11 P(A) = Add the probabilities of the individual events. P(A or E) = P(A) + P(E) 2 11 1 11 3 11 = + = 3 11 The probability of choosing an A or an E is .

  8. 1 8 P(4) = Additional Example 2: Finding the Probability of Disjoint Events Find the probability of each set of disjoint events. B. spinning a 3 or a 4 on a spinner with eight equal sectors numbered 1-8 1 8 P(3) = Add the probabilities of the individual events. P(3 or 4) = P(3) + P(4) 1 8 1 8 2 8 1 4 = + = = 1 4 The probability of choosing a 3 or a 4 is .

  9. 3 10 P(E) = Check It Out! Example 2 Find the probability of each set of disjoint events. A. choosing an I or an E from the letters in the word centimeter 1 10 P(I) = Add the probabilities of the individual events. P(I or E) = P(I) + P(E) 1 10 3 10 4 10 2 5 = + = = 2 5 The probability of choosing an I or an E is .

  10. 1 6 P(4) = Check It Out! Example 2 Find the probability of each set of disjoint events. B. spinning a 2 or a 4 on a spinner with six equal sectors numbered 1-6 1 6 P(2) = Add the probabilities of the individual events. P(2 or 4) = P(2) + P(4) 1 6 1 6 2 6 1 3 = + = = 1 3 The probability of choosing a 2 or a 4 is .

  11. First Roll 4 3 5 6 1 2 5 3 2 6 4 1 7 6 4 7 3 8 2 5 6 9 7 8 5 4 3 9 8 10 7 5 4 6 7 11 8 9 10 5 6 7 10 9 11 6 8 12 Second Roll Additional Example 3: Recreation Application Sharon rolls two number cubes. What is the probability that the sum of the numbers shown on the cubes is 2 or 8? Step 1 Use a grid to find the sample space. The grid shows all possible sums. There are 36 equally likely outcomes in the sample space.

  12. 5 36 P(sum of 8) = Additional Example 3 Continued Sharon rolls two number cubes. What is the probability that the sum of the numbers shown on the cubes is 2 or 8? Step 2 Find the probability of the set of disjoint events. 1 36 P(sum of 2) = P(sum of 2 or sum of 8) = P(sum of 2) + P(sum of 8) 5 36 1 6 1 36 6 36 = + = = 1 6 The probability that the sum of the cubes is 2 or 8 is .

  13. First Roll 4 3 5 6 1 2 5 3 2 6 4 1 7 6 4 7 3 8 2 5 6 9 7 8 5 4 3 9 8 10 7 5 4 6 7 11 8 9 10 5 6 7 10 9 11 6 8 12 Second Roll Check It Out! Example 3 Sun Li rolls two number cubes. What is the probability that the sum of the numbers shown on the cubes is 3 or 4? Step 1 Use a grid to find the sample space. The grid shows all possible sums. There are 36 equally likely outcomes in the sample space.

  14. 3 36 P(sum of 4) = Check It Out! Example 3 Continued Sun Li rolls two number cubes. What is the probability that the sum of the numbers shown on the cubes is 3 or 4? Step 2 Find the probability of the set of disjoint events. 2 36 P(sum of 3) = P(sum of 3 or sum of 4) = P(sum of 3) + P(sum of 4) 2 36 3 36 5 36 = + = 5 36 The probability that the sum of the cubes is 3 or 4 is .

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