Transcript 3.3 The Addition Rule

Section 3.3
The Addition Rule
Larson/Farber Ch. 3
War
Warm Up
1. Nick picks marbles from a jar that contains 3 red,
2 blue, and 5 green marbles. What is the
probability that Nick picks a green marble given
that it was not blue?
2. Jamie picks two cards from a standard deck of
cards (without replacement). What is the
probability that Jamie chooses a queen on her
second pick given that she chose a queen on her
first pick?
Larson/Farber Ch. 3
Objectives/Assignment
• How to determine if two events are
mutually exclusive
• How to use the addition rule to find the
probability of two events.
Larson/Farber Ch. 3
What is different?
• In probability and statistics, the word “or” is
usually used as an “inclusive or” rather
than an “exclusive or.” For instance, there
are three ways for “Event A or B” to occur.
– A occurs and B does not occur
– B occurs and A does not occur
– A and B both occur
Larson/Farber Ch. 3
Independent does not mean mutually exclusive
• Students often confuse the concept of
independent events with the concept of
mutually exclusive events.
Larson/Farber Ch. 3
Study Tip
• By subtracting P(A and B), you avoid
double counting the probability of
outcomes that occur in both A and B.
Larson/Farber Ch. 3
Compare “A and B” to “A or B”
The compound event “A and B” means that A
and B both occur in the same trial. Use the
multiplication rule to find P(A and B).
The compound event “A or B” means either A
can occur without B, B can occur without A or
both A and B can occur. Use the addition rule
to find P(A or B).
B
A
A and B
Larson/Farber Ch. 3
B
A
A or B
Mutually Exclusive Events
Two events, A and B, are mutually exclusive if
they cannot occur in the same trial.
A = A person is under 21 years old
B = A person is running for the U.S. Senate
A = A person was born in Philadelphia
B = A person was born in Houston
A
B
Mutually exclusive
P(A and B) = 0
When event A occurs it excludes event B in the same trial.
Larson/Farber Ch. 3
Non-Mutually Exclusive Events
If two events can occur in the same trial, they are
non-mutually exclusive.
A = A person is under 25 years old
B = A person is a lawyer
A = A person was born in Philadelphia
B = A person watches West Wing on TV
A and B
Non-mutually exclusive
P(A and B) ≠ 0
Larson/Farber Ch. 3
A
B
The Addition Rule
The probability that one or the other of two events will
occur is:
P(A) + P(B) – P(A and B)
A card is drawn from a deck. Find the
probability it is a king or it is red.
A = the card is a king B = the card is red.
P(A) = 4/52 and P(B) = 26/52
but P(A and B) = 2/52
P(A or B) = 4/52 + 26/52 – 2/52
= 28/52 = 0.538
Larson/Farber Ch. 3
The Addition Rule
A card is drawn from a deck. Find the
probability the card is a king or a 10.
A = the card is a king B = the card is a 10.
P(A) = 4/52 and P(B) = 4/52 and P(A and B) = 0/52
P(A or B) = 4/52 + 4/52 – 0/52 = 8/52 = 0.054
When events are mutually exclusive,
P(A or B) = P(A) + P(B)
Larson/Farber Ch. 3
Contingency Table
The results of responses when a sample of adults in
3 cities was asked if they liked a new juice is:
Omaha
Yes
100
No
125
Undecided 75
Total
300
Seattle
150
130
170
450
Miami
150
95
5
250
Total
400
350
250
1000
One of the responses is selected at random. Find:
1. P(Miami and Yes)
3. P(Miami or Yes)
2. P(Miami and Seattle) 4. P(Miami or Seattle)
Larson/Farber Ch. 3
Contingency Table
Yes
No
Undecided
Total
Omaha
100
125
75
300
Seattle
150
130
170
450
Miami
150
95
5
250
Total
400
350
250
1000
One of the responses is selected at random. Find:
1. P(Miami and Yes)
= 250/1000 • 150/250 = 150/1000
= 0.15
2. P(Miami and Seattle) = 0
Larson/Farber Ch. 3
Contingency Table
Yes
No
Undecided
Total
Omaha
100
125
75
300
Seattle
150
130
170
450
Miami
150
95
5
250
Total
400
350
250
1000
3 P(Miami or Yes)
250/1000 + 400/1000 – 150/1000
= 500/1000 = 0.5
4. P(Miami or
Seattle)
250/1000 + 450/1000 – 0/1000
= 700/1000 = 0.7
Larson/Farber Ch. 3
Summary
For complementary events P(E') = 1 - P(E)
Subtract the probability of the event from one.
The probability both of two events occur
P(A and B) = P(A) • P(B|A)
Multiply the probability of the first event by the conditional probability
the second event occurs, given the first occurred.
Probability at least one of two events occur
P(A or B) = P(A) + P(B) - P(A and B)
Add the simple probabilities, but to prevent double counting, don’t
forget to subtract the probability of both occurring.
Larson/Farber Ch. 3