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AP Statistics Notes: 6.3 – General Probability Rules (Page 1)
Objective(s): The learner will be able to use the rules of probability to find probabilites of outcomes.
Rules:
1) 0 ≤ P(A) ≤ 1
2) P(S) = 1
3) Addition Rule: P(A  B) = P(A) + P(B) – P(A ∩ B)
4) Complement Rule: P(Ac) = 1 – P(A)
5) Multiplication Rule: P(A ∩ B) = P(A)•P(B) if A and B are independent
Marginal Probability: the probability of an event happening (row and column total probabilities)
Conditional Probability: P(A | B) = The probability of A happening, given that B has already happened.
P(A|B) =P(A ∩ B)
P(B)
Example 1
Democrat
Republican
Independent
Total
P(B|A) = P(A ∩ B)
P(A)
For Against Total
100 200 300
200 100 300
50 150 200
350 450 800
Marginal Probabilities
P(D) =
Conditional Probabilities
P(D|F) =
P(R) =
P(R|F) =
P(I) =
P(I|A) =
P(F) =
P(F|D) =
P(A) =
P(A|I) =
Example 2: A = likes baseball, B = likes soccer, P(A) = .4, P(B) = .3, P(A ∩ B) = .1
Describe in words and use a table to find the probabilities of:
P(A) =
P(B) =
P(A  B) =
P(A ∩ B) =
P(A | B) =
P(B | A) =
P(Ac) =
P(Bc) =
P(A | Bc) =
P(Ac | B) =
AP Statistics Notes: 6.3 – General Probability Rules (Page 2)
Definitions:
Independence: If A and B are independent, then the result of A has no impact on outcomes of B.
1. P(A ∩ B) = P(A) ∙ P(B)
2. P(A|B) = P(A)
3. P(B|A) = P(B)
Dependence: If A and B are dependent, then the result of A impacts the outcomes of B.
1. P(A ∩ B) = P(A) ∙P(B|A)
2. P(A ∩ B) = P(B) ∙ P(A|B)
EXAMPLE #1: If you throw two fair dice, what is the probability of getting a 5 on each die?
event A in words:
P(A):
event B in words:
P(B):
independent events?:
answer:
EXAMPLE #2: Compute the probability of drawing two aces from a deck of cards if the first card is not
replaced before the second draw.
event A in words:
P(A):
event B in words:
P(B):
independent events?:
answer:
EXAMPLE #3: Compute the probability of drawing two aces from a deck of cards if the first card is
replaced before the second draw.
event A in words:
P(A):
event B in words:
P(B):
independent events?:
answer:
EXAMPLE #4: A quality control procedure for testing flash bulbs consists of drawing two bulbs at
random from each lot of 100 without replacing the first bulb before drawing the second. If both bulbs are
defective, the entire lot is rejected. If the probability that a bulb is defective is 0.01, find the probability:
1. that the lot would not be rejected
2. that the lot would be rejected
AP Statistics Notes: 6.3 – General Probability Rules (Homework - Page 1)
1. A is the event that a person is female. B is the event that a person likes math. If P(A) = 0.6, P(B) = 0.2,
and P(A  B) = 0.1, describe the following events in words and find their probabilities (using a table or a
Venn diagram).
a. P(A)
b. P(B)
c. P(Ac)
d. P(A  B)
e. P(Ac  Bc)
f. P(A | B)
g. P(A  Bc)
h. P(B | A)
i. P(Ac | B)
j. P(Bc | Ac)
k. P(A | Bc)
l. P(Ac  B)
m. P(Ac  Bc)
n. P(A  Bc)
2. A is the event someone likes soccer. B is the event that someone likes reading. If P(A) = 0.2, P(B) = 0.3,
and P(A  B) = 0.4, describe the following events in words and find their probabilities (using a table or a
Venn diagram).
a. P(A)
b. P(B)
c. P(Ac)
d. P(A  B)
e. P(Ac  Bc)
f. P(A | B)
g. P(A  Bc)
h. P(B | A)
i. P(Ac | B)
j. P(Bc | Ac)
k. P(A | Bc)
l. P(Ac  B)
AP Statistics Notes: 6.3 – General Probability Rules (Homework - Page 2)
3. Andrew is 55 and the probability he will be alive in 10 years is 0.72. Ellen is 35 and the probability she
will be alive in 10 years is 0.92. Assuming that the life span of one will have no effect on the life span of
the other:
a. What is the probability they will both be alive in 10 years?
b. What is the probability they will both be dead in 10 years?
c. What is the probability that Andrew will be alive and Ellen dead in 10 years?
d. What is the probability that Ellen will be alive and Andrew dead in 10 years?
4. If you spin a spinner (with the numbers 1-4 equally represented) five times, what is the probability
that:
a. you get all evens
b. you get all threes
c. you get no threes
d. you get at least one three
5. If A & B are independent, P(A|B) = .4, and P(B) = .2, find:
a. P(A)
b. P(Ac)
c. P(A ∩ B)
d. P(A|Bc)
e. P(A  B)
6. If A = draw a heart, and B = draw an ace, find the probability of drawing (no replacement):
a. a heart, then an ace
b. an ace, then a heart
c. two hearts
d. two aces
e. at least one ace
7. If P(A) = .5, and P(A ∩ B) = .3, What is P(B) if A & B are independent?
AP Statistics Notes: 6.3 – General Probability Rules (Homework - Page 3)
8. If A & B are independent what is P(A) if P(A|B) = .4 and P(B) = .2?
9. A class has 60% males. If 20% of the males and 30% of the females got over 1200 on the SAT’s find the
probability that a person:
a) is male
b) got over 1200
c) got over 1200, given she’s female
d) is a female, given that she got more than 1200
e) is a male, given that he got under 1200
f) is a female and got more than 1200
10. You roll a die 4 times. What is the probability that you get: a) all fives, b) no fives, c) at least 1 five?
11. If A & B are independent, P(B|A) = .3, and P(A) = .4, find: P(B), P(A ∩ B), P(A |B), P(A  B), P(Ac  B).
AP Statistics Notes: 6.3 – General Probability Rules (Homework - Page 4)
12. If A & B are independent, P(A ∩B) = 30% and P(A) = 50%, find P(B|A).
13. 10% of the school is 12th graders, 20% is 11th, 30% is 10th, and 40% is 9th. If 80% of the 12th graders,
70% of the 11th, 60% of the 10th, and 50% of the 9th graders pass:
a. What percent of students pass?
b. If you know that a student passed, what is the probability that he/she is a 10th grader?
14. A company retains a psychologist to assess whether job applicants are suited for assembly-line work.
The psychologist classifies applicants as A (well suited), B (marginal), or C (not suited). The company is
concerned about event D: the employee leaves the company within a year of being hired. Data on all
people hired in the past 5 years give these probabilities:
P(A) = 0.4
P(B) = 0.3
P(C) = 0.3
P(A and D) = 0.1
P(B and D) = 0.1
What is the probability that an employee leaves within a year?
15. P(X) = 0.25 and P(Y) = 0.40. If P(X | Y) = 0.20, what is P(Y | X)?
16. A mortality table is given as follows:
Age
# Surviving
0
10,000
20
9,700
40
9,240
60
7,800
80
4,300
What is the probability that a 20-year-old will survive to be 60?
P(C and D) = 0.2