larson/farber ch. 3 section 3.3 the addition rule
TRANSCRIPT
Larson/Farber Ch. 3
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).
A B
A or BA and B
A B
Larson/Farber Ch. 3
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 PhiladelphiaB = A person was born in Houston
A B Mutually exclusiveP(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 oldB = A person is a lawyer
A = A person was born in PhiladelphiaB = A person watches West Wing on TV
A BNon-mutually exclusiveP(A and B) ≠ 0
A and B
Larson/Farber Ch. 3
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) = 2/52P(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
The results of responses when a sample of adults in 3 cities was asked if they liked a new juice is:
Contingency Table
3. P(Miami or Yes)
4. P(Miami or Seattle)
Omaha Seattle Miami TotalYes 100 150 150 400No 125 130 95 350Undecided 75 170 5 250Total 300 450 250 1000
One of the responses is selected at random. Find:
1. P(Miami and Yes)
2. P(Miami and Seattle)
Larson/Farber Ch. 3
Contingency Table
1. P(Miami and Yes)
2. P(Miami and Seattle)
= 250/1000 • 150/250 = 150/1000 = 0.15
= 0
Omaha Seattle Miami TotalYes 100 150 150 400No 125 130 95 350Undecided 75 170 5 250Total 300 450 250 1000
One of the responses is selected at random. Find:
Larson/Farber Ch. 3
Contingency Table
3 P(Miami or Yes)
4. P(Miami or Seattle)
250/1000 + 450/1000 – 0/1000= 700/1000 = 0.7
Omaha Seattle Miami TotalYes 100 150 150 400No 125 130 95 350Undecided 75 170 5 250Total 300 450 250 1000
250/1000 + 400/1000 – 150/1000= 500/1000 = 0.5
Larson/Farber Ch. 3
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.
For complementary events P(E') = 1 - P(E)Subtract the probability of the event from one.
The probability both of two events occurP(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.
Summary