Math 2 Name ______

Lesson 7-3 The Addition Counting Principle, Mutually Exclusive, and Complements

Learning Goals:

·  I can apply the Addition Rule to determine and interpret the probability of the union of two events using the formula P(A or B) = P(A) + P(B) – P(A and B).

·  I can explain and apply the meaning of mutually exclusive events and complementary events

Example 1

A die is rolled. What is the probability that the number is even and less than 4?

Practice 1

1. A die is rolled. What is the probability that the number is prime and less than 5?

2. A set of polygons contains a square, a rectangle, a rhombus and a trapezoid. If one polygon is chosen at random, what is the probability that the polygon has all sides equal in lengthandall right angles? (Hint: A square and a rhombus have all sides equal in length. A square and a rectangle have all right angles.)

3. Two dice are rolled, what is the probability of throwing a 4 on the first die and a 3 on the second die?

Two events that have NO outcomes in common are calledmutually exclusive. These are events that cannot occur at the same time. If two events are mutually exclusive, then P(A and B) = 0

Example 2
A pair of dice is rolled. The events ofrolling a sum of 6and ofrolling a doublehave the outcome (3,3) in common. These two events areNOT mutually exclusive.


Example 3
A pair of dice is rolled. The events ofrolling a sum of 9and rolling a doubleshave

NO outcomes in common. These two eventsARE mutually exclusive.

The rule for “OR” takes into account those values that may get counted more than once when the probability is determined.

Another way to state this is:

The probability of the union of two events is given by:

P(A∪B)= P(A) + P(B)– P(A∩B)

Here, P(A) is the probability of event A, P(B) is the probability of event B.

Also, P(A∩B) is the probability of the intersection of eventsA and B.

Example 4

A die is rolled. What is the probability that the number is evenorless than 4?

(rolling a 2 is in both events)

Answer:Probability = P(A) + P(B) - P(A and B)
= 3/6 + 3/6 - 1/6

=5/6

Practice 2

1. A die is rolled. What is the probability that the number is prime or less than 5?

2. A set of polygons contains a square, a rectangle, a rhombus and a trapezoid. If one polygon is chosen at random, what is the probability that the polygon has all sides equal in lengthorall right angles? (Hint: A square and a rhombus have all sides equal in length. A square and a rectangle have all right angles.)

3. Two dice are rolled, what is the probability of throwing a 4 on the first die or a 3 on the second die?

4. A piggybank contains 2 quarters, 3 dimes, 4 nickels, and 5 pennies. One coin is removed at random. What is the probability that the coin is a dimeora nickel?

IfAis an event within the sample spaceSof an activity or experiment, thecomplementofA(denotedA') consists of all outcomes inSthat arenot inA.
ThecomplementofAis everything else in the problem that is NOT inA.

Consider these experiments where an event and its complement are shown:

Experiment: Tossing a coin
Event / A / The coin shows heads.
Complement / A' / The coin shows tails.
Experiment: Drawing a card
Event / A / The card is black.
Complement / A' / The card is red.

The probability of the complement of an event is one minus the probability of the event.

Complement:

Example 5

Use the complement to find the probability of rolling two dice and getting a sum of less than 11.

Practice 3

1. One die is rolled. Two possible events are rolling a number less than 5 and rolling a number which is a multiple of 5. Are these two events mutually exclusive? Explain

2. A pair of dice is rolled. Two possible events are rolling a sum that is greater than 8 and rolling a sum that is an even number. Are these two events mutually exclusive events? Explain.

3. A pair of dice is rolled. A possible event is rolling a sum that is a multiple of 5. What is the complement of this event? Find the probability of the complement of this event.

4. A pair of dice is rolled. Two possible events are rolling a sum which is a multiple of 3 and rolling a number which is a multiple of 5. Are these two events mutually exclusive? Explain

5. A pair of dice is rolled and the resulting sum is odd. What is the complement of this event? What is the probability of the complement?

6. A bag contains marbles of three different colors: 5 black, 6 white, and 5 red. Three marbles are selected at random, without replacement. Find the probability that the selection contains each of the outcomes listed below. Express the answer as a decimal to thenearest thousandth.

a.)  three black marbles


b.) three white marbles


c.) one white marble followed by two
black marbles


d.) a red, a black and a white marble in that order