P(2)
=
1
6
P(5)
=
1
6
P(2 or 5)
=
P(2)
+
P(5)
=
1
+
1
6
6
=
2
6
=
1
3
Addition Rules for Probability
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 | ||||
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P(5) | = | 1 |
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6 |
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P(2 or 5) | = | P(2) | + | P(5) |
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| = | 1 | + | 1 |
6 | 6 | |||
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= | 2 |
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6 |
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= | 1 |
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3 |
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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 | ||||
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P(blue) | = | 1 |
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4 |
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P(red or blue) | = | P(red) | + | P(blue) |
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| = | 1 | + | 1 |
4 | 4 | |||
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= | 2 |
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4 |
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= | 1 |
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2 |
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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 | ||||
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P(green) | = | 3 |
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10 |
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P(yellow or green) | = | P(yellow) | + | P(green) |
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| = | 4 | + | 3 |
10 | 10 | |||
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= | 7 |
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10 |
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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) |
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| = | 4 | + | 13 | - | 1 |
52 | 52 | 52 | ||||
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= | 16 |
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52 |
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= | 4 |
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13 |
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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.
Additional 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 | ||||
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= | 17 |
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30 |
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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) |
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| = | 0.32 | + | 0.09 | - | 0.15 |
| = | 0.26 |
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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)
