Exam 5: Probabilistic Thinking

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?

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?

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?

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

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:

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:

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?

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:

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:

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:

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:

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:

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.

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:

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?

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

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?

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?

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):

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: