Exam 5: Probabilistic Thinking

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?

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?

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?

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

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.

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?

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

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?

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

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?

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?

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?

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?

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?