AP Study Session

1
AP Study Session

• Probability

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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 Barrons 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 Barrons 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 Barrons 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) doesnt equal P(hurricane water damage)

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Adapted from Barrons 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 Barrons 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 Barrons 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 Barrons 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 Barrons 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 Barrons 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 Barrons 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?

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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-Barrons 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

• Barrons 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.

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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(Xgt700) .1332, P(X is between 700 and
  740) .1105
• P(Xlt740 given that Xgt700) .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.

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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.

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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?

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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

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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.

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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).

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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.

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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 cant 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.