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 Addition Rules for Probability Unit 6 > Lesson 6 of 12

Experiment 1:   A single 6-sided die is rolled. What is the probability of rolling a 2 or a 5?
Possibilities:
 1 The number rolled can be a 2. 2 The number rolled can be a 5.
Events:   These events are mutually exclusive since they cannot occur at the same time.
Probabilities:   How do we find the probabilities of these mutually exclusive events? We need a rule to guide us.

 Addition Rule 1: When two events, A and B, are mutually exclusive, the probability that A or B will occur is the sum of the probability of each event. P(A or B) = P(A) + P(B)

Let's use this addition rule to find the probability for Experiment 1.

Experiment 1:   A single 6-sided die is rolled. What is the probability of rolling a 2 or a 5?
Probabilities:
 P(2) = 1 6 P(5) = 1 6 P(2 or 5) = P(2) + P(5) = 1 + 1 6 6 = 2 6 = 1 3

Experiment 2:   A spinner has 4 equal sectors colored yellow, blue, green, and red. What is the probability of landing on red or blue after spinning this spinner?
Probabilities:
 P(red) = 1 4 P(blue) = 1 4 P(red or blue) = P(red) + P(blue) = 1 + 1 4 4 = 2 4 = 1 2

Experiment 3:   A glass jar contains 1 red, 3 green, 2 blue, and 4 yellow marbles. If a single marble is chosen at random from the jar, what is the probability that it is yellow or green?
Probabilities:
 P(yellow) = 4 10 P(green) = 3 10 P(yellow or green) = P(yellow) + P(green) = 4 + 3 10 10 = 7 10

 In each of the three experiments above, the events are mutually exclusive. Let's look at some experiments in which the events are non-mutually exclusive.

Experiment 4:   A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a king or a club?
Probabilities:
 P(king or club) = P(king) + P(club) - P(king of clubs) = 4 + 13 - 1 52 52 52 = 16 52 = 4 13
 In Experiment 4, the events are non-mutually exclusive. The addition causes the king of clubs to be counted twice, so its probability must be subtracted. When two events are non-mutually exclusive, a different addition rule must be used.

 Addition Rule 2: When two events, A and B, are non-mutually exclusive, the probability that A or B will occur is: P(A or B) = P(A) + P(B) - P(A and B)

 In the rule above, P(A and B) refers to the overlap of the two events. Let's apply this rule to some other experiments.

Experiment 5:   In a math class of 30 students, 17 are boys and 13 are girls. On a unit test, 4 boys and 5 girls made an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student?
Probabilities:
 P(girl or A) = P(girl) + P(A) - P(girl and A) = 13 + 9 - 5 30 30 30 = 17 30

 Experiment 6: On New Year's Eve, the probability of a person having a car accident is 0.09. The probability of a person driving while intoxicated is 0.32 and probability of a person having a car accident while intoxicated is 0.15. What is the probability of a person driving while intoxicated or having a car accident? Probabilities:
 P(intoxicated or accident) = P(intoxicated) + P(accident) - P(intoxicated and accident) = 0.32 + 0.09 - 0.15 = 0.26

Summary:   To find the probability of event A or B, we must first determine whether the events are mutually exclusive or non-mutually exclusive. Then we can apply the appropriate Addition Rule:

 Addition Rule 1: When two events, A and B, are mutually exclusive, the probability that A or B will occur is the sum of the probability of each event. P(A or B) = P(A) + P(B) Addition Rule 2:: When two events, A and B, are non-mutually exclusive, there is some overlap between these events. The probability that A or B will occur is the sum of the probability of each event, minus the probability of the overlap. P(A or B) = P(A) + P(B) - P(A and B)

### Exercises

 Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, choose a different button.

 1. A day of the week is chosen at random. What is the probability of choosing a Monday or Tuesday? None of the above. RESULTS BOX:

 2. In a pet store, there are 6 puppies, 9 kittens, 4 gerbils and 7 parakeets. If a pet is chosen at random, what is the probability of choosing a puppy or a parakeet? 1 None of the above. RESULTS BOX:

 3. The probability of a New York teenager owning a skateboard is 0.37, of owning a bicycle is 0.81 and of owning both is 0.36. If a New York teenager is chosen at random, what is the probability that the teenager owns a skateboard or a bicycle? 1.18 0.7 0.82 None of the above. RESULTS BOX:

 4. A number from 1 to 10 is chosen at random. What is the probability of choosing a 5 or an even number? All of the above. RESULTS BOX:

 5. A single 6-sided die is rolled. What is the probability of rolling a number greater than 3 or an even number? 1 None of the above. RESULTS BOX:

### Unit 6: Probability Theory

 Lesson Introduction to Probability Certain and Impossible Events Sample Spaces The Complement of an Event Mutually Exclusive Events Addition Rules for Probability Independent Events Dependent Events Conditional Probability Practice Exercises Challenge Exercises Solutions

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