Experiment: 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.
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 | ||||
= | 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? |
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? |
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? |
4. | A number from 1 to 10 is chosen at random. What is the probability of choosing a 5 or an even number? |
5. | A single 6-sided die is rolled. What is the probability of rolling a number greater than 3 or an even number? |