1) The document discusses key concepts in probability including independence, mutually exclusive events, and normal distributions. 2) It provides examples of probability calculations including finding the probability of independent vs. dependent events, P(A union B) for mutually exclusive vs. independent vs. subset events, and the probability of scores below a cutoff for a normal distribution. 3) It also covers the binomial probability model and solving a multi-part probability word problem involving sampling without replacement to determine gender ratios.

1. AP Study Session
Probability

2. Independence
If knowing that Event A has occurred
gives you information about Event B,
then Events A & B are not independent.
Ex. Football players and knee
problems
Ex. Outcome on 2 fair coins

3. Mutually Exclusive Events
Also called disjoint. They have no outcomes
in common.
Knowing that Event A has occurred does give
you information about B. If A occurred, then
B could not have occurred. Thus, disjoint
events are not independent.
The P(A and B) when A & B are disjoint is 0.

4. Adapted from Barron’s p. 394 #7
Suppose that for a certain Caribbean
island in any 3 year period the
probability of a major hurricane is .25,
the probability of water damage is .44,
and the probability of both a hurricane
and water damage is .22.
A Venn diagram helps to organize
information.

5. Adapted from Barron’s p. 394 #7
Suppose that for a certain Caribbean
island in any 3 year period the
probability of a major hurricane is .25,
the probability of water damage is .44,
and the probability of both a hurricane
and water damage is .22.
Are the events hurricane and water
damage independent?

6. Adapted from Barron’s p. 394 #7
Suppose that for a certain Caribbean island in
any 3 year period the probability of a major
hurricane is .25, the probability of water
damage is .44, and the probability of both a
hurricane and water damage is .22.
Are the events hurricane and water damage
independent? No, because P(hurricane) x
P(water damage) doesn’t equal P(hurricane &
water damage)

7. Adapted from Barron’s p. 394 #7
Suppose that for a certain Caribbean
island in any 3 year period the
probability of a major hurricane is .25,
the probability of water damage is .44,
and the probability of both a hurricane
and water damage is .22.
What is the probability of water damage
given that there is a hurricane?

8. Adapted from Barron’s p. 394 #7
Suppose that for a certain Caribbean island in
any 3 year period the probability of a major
hurricane is .25, the probability of water
damage is .44, and the probability of both a
hurricane and water damage is .22.
What is the probability of water damage given
that there is a hurricane? Answer: .22/.25
= .88

9. Adapted from Barron’s p. 369 #10
Given the probabilities P(A) = .3 and
P(B) = .2, what is the P(A union B) if A
and B are mutually exclusive?

10. Adapted from Barron’s p. 369 #10
Given the probabilities P(A) = .3 and
P(B) = .2, what is the P(A union B) if A
and B are mutually exclusive?
Answer: .3 + .2 = .5

11. Adapted from Barron’s p. 369 #10
Given the probabilities P(A) = .3 and
P(B) = .2, what is the P(A union B) if A
and B are independent?

12. Adapted from Barron’s p. 369 #10
Given the probabilities P(A) = .3 and
P(B) = .2, what is the P(A union B) if A
and B are independent?
Answer: .3 + .2 - .06 =.44

13. Adapted from Barron’s p. 369 #10
Given the probabilities P(A) = .3 and
P(B) = .2, what is the P(A union B) if B
is a subset of A?
Answer: .3

14. Tree Diagram
A plumbing contractor obtains 60% of
her boiler circulators from a company
whose defect rate is .005, and the rest
from a company whose defect rate is
0.010. What proportion of the
circulators can be expected to be
defective?

15. Tree Diagram
A plumbing contractor obtains 60% of
her boiler circulators from a company
whose defect rate is .005, and the rest
from a company whose defect rate is
0.010. What proportion of the
circulators can be expected to be
defective?
Answer: (.6)(.005) + (.4)(.010) = .007

16. Tree Diagram-Barron’s P. 368 #6
A plumbing contractor obtains 60% of
her boiler circulators from a company
whose defect rate is .005, and the rest
from a company whose defect rate is
0.010. If a circulator is defective,what is
the probability that it came from the first
company?

17. Tree Diagram
A plumbing contractor obtains 60% of
her boiler circulators from a company
whose defect rate is .005, and the rest
from a company whose defect rate is
0.010. If a circulator is defective,what is
the probability that it came from the first
company?
Answer: (.6)(.005)/.007

18. Probability with Normal
Distributions
Barron’s p. 373 #31
The mean Law School Aptitude Test (LSAT)
score for applicants to a particular law is 650
with a standard deviation of 45. Suppose that
only applicants with scores above 700 are
considered. What percentage of the
applicants considered have scores below
740? Assume the scores are normally
distributed.

19. Probability with Normal
Distributions
The mean Law School Aptitude Test (LSAT)
score for applicants to a particular law is 650
with a standard deviation of 45. Suppose that
only applicants with scores above 700 are
considered. What percentage of the
applicants considered have scores below
740? Assume the scores are normally
distributed.
Answer: P(X>700) = .1332, P(X is between 700
and 740) = .1105
P(X<740 given that X>700) = .1105/.1332 = .
8297

20. Binomial Probability
Remember BINS
B-Binary
I-Independent trials
N-Set number of trials
S-Probability of success is the same for
each trial

21. AP Free Response 2004 #3
At an archaeological site that was an ancient swamp,
the bones from 20 brontosaur skeletons have been
unearthed. The bones do not show any sign of
disease or malformation. It is thought that these
animals wandered into a deep area of the swamp and
became trapped in the swamp bottom. The 20 left
femur bones (thigh bones) were located and 4 of
these left femurs are to be randomly selected without
replacement for DNA testing to determine gender.
A) Let X be the number out of the 4 selected left
femurs that are from males. Based on how these
bones were sampled, explain why the probability
distribution of X is not binomial.

22. AP Free Response 2004 #3
A) Let X be the number out of the 4 selected
left femurs that are from males. Based on
how these bones were sampled, explain why
the probability distribution of X is not binomial.
Answer: X is not binomial since the trials are
not independent and the conditional
probabilities of selecting a male change at
each trial depending on the previous
outcome(s), due to the sampling without
replacement.

23. AP Free Response 2004 #3
At an archaeological site that was an ancient swamp,
the bones from 20 brontosaur skeletons have been
unearthed. The bones do not show any sign of
disease or malformation. It is thought that these
animals wandered into a deep area of the swamp and
became trapped in the swamp bottom. The 20 left
femur bones (thigh bones) were located and 4 of
these left femurs are to be randomly selected without
replacement for DNA testing to determine gender.
B) Suppose that the group of 20 brontosaurs whose
remains were found in the swamp had been made up
of 10 males and 10 females. What is the probability
that all 4 in the sample to be tested are male?

24. AP Free Response 2004 #3
B) Suppose that the group of 20
brontosaurs whose remains were found
in the swamp had been made up of 10
males and 10 females. What is the
probability that all 4 in the sample to be
tested are male?
Answer: (10/20)(9/19)(8/18)(7/17) = .
043

25. AP Free Response 2004 #3
At an archaeological site that was an ancient swamp,
the bones from 20 brontosaur skeletons have been
unearthed. The bones do not show any sign of
disease or malformation. It is thought that these
animals wandered into a deep area of the swamp and
became trapped in the swamp bottom. The 20 left
femur bones (thigh bones) were located and 4 of
these left femurs are to be randomly selected without
replacement for DNA testing to determine gender.
C) The DNA testing revealed that all 4 femurs tested
were from males. Based on this result and your
answer from part (b), do you think that males and
females were equally represented in the group of 20
brontosaurs stuck in the swamp? Explain.

26. AP Free Response 2004 #3
C) The DNA testing revealed that all 4 femurs tested
were from males. Based on this result and your
answer from part (b), do you think that males and
females were equally represented in the group of 20
brontosaurs stuck in the swamp? Explain.
Answer: No. If males and females were equally
represented, the probability of observing four males
is small (0.043).

27. AP Free Response 2004 #3
At an archaeological site that was an ancient swamp,
the bones from 20 brontosaur skeletons have been
unearthed. The bones do not show any sign of
disease or malformation. It is thought that these
animals wandered into a deep area of the swamp and
became trapped in the swamp bottom. The 20 left
femur bones (thigh bones) were located and 4 of
these left femurs are to be randomly selected without
replacement for DNA testing to determine gender.
D) Is it reasonable to generalize your conclusion in
part (c) pertaining to the group of 20 brontosaurs to
the population of all brontosaurs? Explain why or
why not.

28. AP Free Response 2004 #3
D) Is it reasonable to generalize your
conclusion in part (c) pertaining to the group
of 20 brontosaurs to the population of all
brontosaurs? Explain why or why not.
Answer: No, we can’t generalize to the
population of all brontosaurs because it is not
reasonable to regard this sample as a
random sample from the population of all
brontosaurs; there is reason to suspect that
this sampling method might cause bias.