Addition Rules for Probability
| 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)| | |
| > From <http://www.mathgoodies.com/lessons/vol6/addition_rules.html> |
