Deck 5: Probabilistic Thinking

Question 1

Caroline has just learned that she needs to be in Washington DC the next morning for an urgent work related matter. She calls United, Delta and American. The early morning flights are all pretty full. Caroline finds out that there is a 25% chance that she can get on a United flight; 20% chance that she can get on Delta and 30% chance that she can get on American. What is the approximate probability that Caroline will be able to get on a plane to Washington DC?

A) 58%.
B) 65%.
C) 42%.
D) 48%.

Answer 1

58%.

Question 2

Consider two boxes: A and B. Box A contains 3 Red and 1 Blue marble. Box B contains 3 Blue and 1 Red marble. Simon tosses a fair die. If the die comes up with the numbers 1 - 3 then he will choose Box A, if it comes up with 4 - 6, then he chooses Box B. Once a box has been chosen (and each box has ½ chance of being chosen), he will reach inside and randomly pick one of the four marbles in that box. Suppose, Simon has picked a red marble. What is the probability that the red marble came from Box B?

A) 1/4.
B) 1/2.
C) 3/4.
D) 1/8.

Answer 2

1/4.

Question 3

Lisa is 35 and pregnant for the first time. As a "mature" mother, she is worried about the possibility of having a baby with Down Syndrome. Chances of a 35-year old mother having a baby with Down Syndrome is 1 in 300. Just to be on the safe side, Lisa decides to have a maternal blood serum test to check for chromosomal abnormalities. If the baby has Down Syndrome, then the blood serum test will deliver a positive result 90% of the time. There is, however, a small "false positive" rate of 7%. Lisa's blood serum test provides a positive result for Down Syndrome. The prior probability that Lisa's child will have Down Syndrome is:

A) 0.003 or 0.3%.
B) 0.037 or 3.7%.
C) 0.357 or 35.7%
D) 0.148 or 14.8%.

Answer 3

0.003 or 0.3%.

Question 4

Lisa is 35 and pregnant for the first time. As a "mature" mother, she is worried about the possibility of having a baby with Down Syndrome. Chances of a 35-year old mother having a baby with Down Syndrome is 1 in 300. Just to be on the safe side, Lisa decides to have a maternal blood serum test to check for chromosomal abnormalities. If the baby has Down Syndrome, then the blood serum test will deliver a positive result 90% of the time. There is, however, a small "false positive" rate of 7%. Lisa's blood serum test provides a positive result for Down Syndrome. The posterior probability that Lisa's child will have Down Syndrome is:

A) 0.037 or 3.7%.
B) 0.003 or 0.3%.
C) 0.03 or 3%.
D) 0.9 or 90%.

Question 5

The functioning of a particular machine is dependent on two safety valves: A and B. The machine will operate fine as long as *one* of the safety valves is functioning properly. Both valves have a 10% chance of failure each. What is the chance that the machine will not break down?

A) 99%.
B) 90%.
C) 95%.
D) 80%.

Question 6

Karishma has just learned that she needs to fly from Auckland to Wellington the next morning for an urgent work-related matter. She calls Air New Zealand and JetStar looking for a ticket. The early morning flights are all pretty full and she learns that there is a 25% chance that she can get on an Air NZ flight and 35% chance that she can get on JetStar. What is the approximate probability that Karishma will be able to get on a plane to Wellington?

A) 51%
B) 35%
C) 65%
D) 25%

Question 7

Lisa is 40 and pregnant for the first time. As a "mature" mother, she is worried about the possibility of having a baby with Down Syndrome. Chances of a 40-year old mother having a baby with Down Syndrome is 1 in 100. Just to be on the safe side, Lisa decides to have a maternal blood serum test to check for chromosomal abnormalities. If the baby has Down Syndrome, then the blood serum test will deliver a positive result 86% of the time. There is, however, a small "false positive" rate of 5%. Lisa's blood serum test provides a positive result for Down Syndrome. The prior probability that Lisa's child will have Down Syndrome is:

A) 86%
B) 1%
C) 5%
D) 63%

Question 8

Lisa is 40 and pregnant for the first time. As a "mature" mother, she is worried about the possibility of having a baby with Down Syndrome. Chances of a 40-year old mother having a baby with Down Syndrome is 1 in 100. Just to be on the safe side, Lisa decides to have a maternal blood serum test to check for chromosomal abnormalities. If the baby has Down Syndrome, then the blood serum test will deliver a positive result 86% of the time. There is, however, a small "false positive" rate of 5%. Lisa's blood serum test provides a positive result for Down Syndrome. The posterior probability that Lisa's child will have Down Syndrome is:

A) 86%.
B) 1%.
C) 5%.
D) 14.8

Question 9

Kookaburra produces its cricket balls in three different sites: S1, S2 and S3. S1 provides 20%, S2 provides 30% and S3 provides 50% of all balls. However, S1 produces 5% (0.05) defective items, S2 produces 2% (0.02) while S3 produces 1% (0.01). If an item sampled at random was found to be defective, what is the POSTERIOR probability it came from S3?

A) 50%.
B) 47.6%.
C) 28.6%.
D) 23.8%.

Question 10

Kookaburra produces its cricket balls in three different sites: S1, S2 and S3. S1 provides 20%, S2 provides 30% and S3 provides 50% of all balls. However, S1 produces 5% (0.05) defective items, S2 produces 2% (0.02) while S3 produces 1% (0.01). If an item sampled at random was found to be defective, what is the POSTERIOR probability it came from S1?

A) 50%.
B) 47.6%.
C) 28.6%.
D) 23.8%.

Question 11

Kookaburra produces its cricket balls in three different sites: S1, S2 and S3. S1 provides 20%, S2 provides 30% and S3 provides 50% of all balls. However, S1 produces 5% (0.05) defective items, S2 produces 2% (0.02) while S3 produces 1% (0.01). If an item sampled at random was found to be defective, what is the POSTERIOR probability it came from S2?

A) 50%.
B) 47.6%.
C) 28.6%.
D) 23.8%.

Question 12

The functioning of a particular machine is dependent on two safety valves: A and B. The machine will operate fine as long as one of the safety valves is functioning properly. Valve A has a 10% chance of failure while valve B has a 5% chance of failure. What is the probability that the machine will NOT break down?

A) 92%.
B) 95%.
C) 99.5%.
D) 6%.

Question 13

The functioning of a particular machine is dependent on two safety valves: A and B. In order for the machine to operate perfectly BOTH valves must function well. Valve A has a 10% chance of failure while valve B has a 5% chance of failure. If you are in charge of maintaining this machine, then you can say that there is at worst a _____ chance of this machine breaking down and so with _____ chance the machine will work fine.

A) 10%; 90%
B) 15.5%; 84.5%
C) 1%; 99%.
D) 90%; 10%.

Question 14

The functioning of a particular machine is dependent on two safety valves: A and B. In order for the machine to operate perfectly BOTH valves must function well. The valves have been known to fail when subjected to extremely stressful conditions such as extreme heat or cold. Valve A has a 10% chance of failure while valve B has a 5% chance of failure under similar stress such as extreme heat or cold. If you are in charge of maintaining this machine, then you can say that there is at worst a _____ chance of this machine breaking down and so with _____ chance the machine will work fine.

A) 10%; 90%.
B) 15%; 85%.
C) 1%; 99%.
D) 90%; 10%.

Question 15

Items used in production process in a factory come from three suppliers S1, S2 and S3. S1 provides 20%, S2 provides 30% and S3 provides 50%. However, S1 produces 5% (0.05) defective items, S2 produces 2% (0.02) while S3 produces 1% (0.01). If an item sampled at random was found to be defective, what is the prior probability it came from Supplier S1?

A) 20%.
B) 5%.
C) 2%.
D) 1%.

Question 16

Items used in production process in a factory come from three suppliers S1, S2 and S3. S1 provides 20%, S2 provides 30% and S3 provides 50%. However, S1 produces 5% (0.05) defective items, S2 produces 2% (0.02) while S3 produces 1% (0.01). If an item sampled at random was found to be defective, what is the approx. posterior probability it came from Supplier S1?

A) 48%.
B) 20%.
C) 5%.
D) 21%.

Question 17

Items used in production process in a factory come from three suppliers S1, S2 and S3. S1 provides 20%, S2 provides 30% and S3 provides 50%. However, S1 produces 5% (0.05) defective items, S2 produces 2% (0.02) while S3 produces 1% (0.01). If an item sampled at random was found to be defective, what is the approx. POSTERIOR probability it came from Supplier S3?

A) 28%.
B) 48%.
C) 24%.
D) 50%.

Question 18

Suppose I asked Simon which one is bigger: the set of 7 letter words ending in "ING" or the set of 7 letter words with "N" as the 6th letter. If Simon responds that the former set is bigger than the latter set, then this would be an example of:

A) Priming.
B) Framing
C) Framing.
D) Conjunction fallacy.

Question 19

Simon has two urns, 1 and 2. Urn 1 contains two white and 1 black pawn. Urn 2 contains 1 white and 2 black pawns. Simon tosses a regular 6-sided die. If the die comes up with 1 through 4, Simon chooses Urn 1, while if the die comes up with 5 or 6, he chooses Urn 2. Once an urn is chosen, he reaches in and randomly picks out a pawn from that urn. Suppose Simon picks a black pawn. What is the probability that the black pawn came from Urn 1?

A) 1/9.
B) 1/2.
C) 1/3.
D) 2/3.

Question 20

Which of the following would be an example of the "conjunction fallacy"?

A) Stating that the set of 7 letter words ending in "ING" is larger than the set of 7 letter words with "N" as the 6th letter.
B) Slowing down well below the speed limit when driving at the sight of police car.
C) Contributions to retirement saving account increase when a company moves from an opt-in to an opt-out savings plan.
D) Asking for a higher selling price for a good one already possesses compared to the price one is willing to pay for buying the same exact good.

Question 21

Linda is 22 and pregnant for the first time. Chances of a 22-year old mother having a baby with Down Syndrome is 1 in 1000. Just to be on the safe side, Linda decides to have a maternal blood serum test to check for chromosomal abnormalities. If the baby has Down Syndrome, then the blood serum test will deliver a positive result 86% of the time. There is, however, a small "false positive" rate of 5%. Linda's blood serum test provides a positive result for Down Syndrome. The probability that Linda's child will have Down Syndrome is (approximately):

A) 86%.
B) 1.7%.
C) 0.1%.
D) 63%.

Question 22

Ceejay saw the following newspaper headline: "Excessive videogame playing linked to heightened risk of depression in young adolescents." Ceejay should be thinking:

A) Videogame playing causes depression in young adolescents.
B) Depression causes young adolescents to play more video games.
C) Videogames are detrimental to mental health and should be severely restricted, if not banned outright.
D) There is most likely another omitted variable at work here that possibly explains incidents of depression and excessive game playing among some young adolescents.

Question 23

A recent article published in the New York Times was titled "Happy children do chores". From this title, we should infer that:

A) There is quite possibly a third omitted variable relating to the nature of some families, where children are happy and do household chores.
B) Happiness causes children to do chores.
C) Children should be forced to do more chores around the house.
D) Doing chores causes children to be happy.

Question 24

Consider two boxes: A and B. Box A contains 8 Red and 2 Blue marbles. Box B contains 2 Red and 8 Blue Red marbles. Jack tosses a fair die. If the die comes up with the numbers 1 or 2, then he will choose Box A, if it comes up with 3, 4, 5 or 6, then he chooses Box B. Once a box has been chosen, he will reach inside and randomly pick one of the ten marbles in that box. Suppose, Jack has picked a red marble. What is the probability that the red marble came from Box B?

A) 1/3.
B) 1/4.
C) 1/2.
D) 1/5.

Question 25

Consider two boxes: A and B. Box A contains 3 Red and 1 Blue marble. Box B contains 3 Blue and 1 Red marble. Bridget tosses a fair die. If the die comes up with the numbers 1 - 4 then she will choose Box A, if it comes up with 5 or 6, then she chooses Box B. Once a box has been chosen, she will reach inside and randomly pick one of the four marbles in that box. Suppose Bridget shows you that she has picked a red marble. What is the probability that the red marble came from Box B?

A) 1/7.
B) 1/3.
C) 3/4.
D) 1/2.

Question 26

When asked people often list war and terrorism as the leading causes of death worldwide. This turns out to be incorrect and most deaths around the world are caused by respiratory illnesses. The fact that people make this mistake can be attributed to:

A) Availability bias.
B) Conjunction Fallacy.
C) Mistaking correlation for causation.
D) Failure to understand the difference between conjunctive and disjunctive events.

Question 27

Most professional athletes tend to perform along the lines of their lifetime averages. If one player performs significantly better in one game or in one season then the chances are that he/she will perform worse in the next game or next season. This phenomenon is usually referred to as:

A) Regression to the mean.
B) Correlation.
C) Conjunctive fallacy.
D) "Hot hands" syndrome.

Question 28

Sports fans and even athletes, particularly in basketball, seem to believe in the so-called "hot hands" syndrome. This belief appears to originate from the fact that people:

A) Often fail to realize that small observations are more likely to throw up more extreme outcomes.
B) Understand the phenomenon of "regression to the mean".
C) Fail to understand the difference between correlation and causation.
D) Fail to understand the difference between conjunctive and disjunctive events.

Question 29

The fact that we do not immediately pick up on the fact that a large country like China may have a small proportion of rich people but still have a very large number of rich can be attributed to:

A) A failure to understand base rates.
B) The conjunction fallacy.
C) The failure to distinguish between correlation and causation.
D) The failure to distinguish between conjunctive and disjunctive events.
Short essay type questions

Question 30

Sarah Jessica Parker is 39. As a "mature" mother, she is worried about the possibility of having a baby with Down Syndrome. Chances of a 39-year old mother having a baby with Down Syndrome is approximately 1 in 100. Just to be on the safe side, Sarah decides to have a maternal blood serum test to check for chromosomal abnormalities. If the baby has Down Syndrome, then the blood serum test will deliver a positive result 90% of the time. There is, however, a small "false positive" rate of 7%. I want you to explain all of this using simple numbers that any layperson can get his/her head around.
Suppose 1000 39-year old mothers choose to get the maternal blood serum test. (a) How many mothers will get a "true positive", i.e. the baby has DS and the test returns a positive result? (b) How many will get a "false positive", i.e. the baby does not have DS, but the test returns a positive result? (c) How many mothers will get a "false negative", i.e. the baby has DS, but the test returns a negative result? (d) Suppose Sarah gets a positive test result. Using only the numbers you have found in the previous parts of this question, what is the posterior probability that her baby has Down Syndrome? How much larger is this compared to her prior probability?

Question 31

Lisa is 40 and pregnant for the first time. As a "mature" mother, she is worried about the possibility of having a baby with Down Syndrome. Chances of a 40-year old mother having a baby with Down Syndrome is 1 in 100. Just to be on the safe side, Lisa decides to have a maternal blood serum test to check for chromosomal abnormalities. If the baby has Down Syndrome, then the blood serum test will deliver a positive result 86% of the time. There is, however, a small "false positive" rate of 5%. Lisa's blood serum test provides a positive result for Down Syndrome. What is the prior probability that Lisa's child will have Down Syndrome? What is the posterior probability that Lisa's child will have Down Syndrome?

Question 32

Items used in production process in a factory come from three suppliers S1, S2 and S3. S1 provides 20%, S2 provides 30% and S3 provides 50%. However, S1 produces 5% (0.05) defective items, S2 produces 2% (0.02) while S3 produces 1% (0.01). If an item sampled at random was found to be defective, what is the prior probability that it came from Supplier S3? What is the posterior probability that it came from Supplier S3?

Question 33

A cab company was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. 85% of the cabs in the city are green, while 15% are blue. A witness identified the cab as Blue. The court tested the reliability of the witness under the circumstances that existed on that night or the accident. The witness correctly identified the true colour 80% of the time, and failed 20% of the time. What is the prior probability that the cab involved in the accident is actually Blue? Following the test administered to the eyewitness, what is the posterior probability that the cab involved in the accident is actually Blue?

Question 34

What is meant by "regression to the mean"? Provide a simple example to illustrate your answer.

Question 35

Why do birth-rates tend to be higher in poorer countries? Explain briefly.