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?
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?
Interactive Math Goodies Software
![[IMAGE]](images/marbles_jar.gif)
![[IMAGE]](images/poker_hand.gif)
![[IMAGE]](images/Agrade_girl.gif)
![[IMAGE]](images/car_crash.gif)
