PROBLEMS ON CONFUSING ASPECTS OF PROBABILITY

1. Using Venn Diagrams

Suppose you are studying a sample of 200 butterflies taken from your garden. You notice

that 40 of them are Orange and 90 of them have Blue spots close to the eye. Upon

examining the Orange ones, you find that 3/4 of them have Blue spots.

Draw a Venn diagram, approximately to scale, representing the above data. For each of the

following questions about this sample, give a numerical value where possible. If it is not

possible to compute the answer exactly, does the data provided allow you to at least

estimate it?

If so, put in your estimate. If not, say why not.

Connect any pairs of probabilities that are inverses of one another with brackets.

a) P(O) = 40/200

b) P(B) = 90/200

c) P(O/B) = 30/90

d) P(B/O) = 30/40

e) P(not-O) = 160/200

f) P(not-B) = 110/200

g) P(not-O/B) = 60/90

h) P(not-B/O) = 10/40

i) P(O/not-B) = 10/110

j) P(B/not-O) = 60/160

Pairs of inverses are: c and d, g and j, h and i.

2. Conjunction

Consider the following description of Bill and then some candidate statements about him.

You do not need to rank order the options yourself. Instead, first specify any limitations the

rules of probability would place on the rankings that you should assign if you were following

the dictates of rationality. Secondly, describe what you would expect most people to actually

do in comparison with what you have specified above.

"Bill is 34 years old. He is intelligent, but unimaginative, compulsive, and generally lifeless. In

school, he was strong in mathematics but weak in social studies and humanities.

Please rank order the following statements by their probability, using 1 for the most probable

and 8 for the least probable.

Bill is a physician who plays poker for a hobby

Bill is an architect

Bill is an accountant

Bill plays jazz for a hobby

Bill surfs for a hobby

Bill is a reporter

Bill is an accountant who plays jazz for a hobby

Bill climbs mountains for a hobby"

Answer: According to the rules of probability, compound events should be less likely than the single events which they are made up of. For example, from the description of Bill, the probability that he is an accountant is very high, while the probability that he plays jazz for a hobby is rather low. Thus, the conjunction of “Bill is an accountant who plays jazz for a hobby” should be ranked lower than both “Bill is an accountant” and “Bill plays jazz for a hobby.” However, most people will rank the conjunction “Bill is an accountant who plays jazz for a hobby” between “Bill is an accountant” and “Bill plays jazz for a hobby.”

3. Bill again

Suppose you now learn that the little description of Bill was drawn at random from a file

drawer that contains the combined notes of a Human Resources clerk who was

interviewing people for two jobs, one in accounting and one in sales. Most of the candidates

(75%) were looking for the sales job but a quarter of them wanted to be accountants.

The clerk describes Bill as follows:

"Bill is 34 years old. He seems to be intelligent, but unimaginative, compulsive, and generally

lifeless. In school, he was strong in mathematics but weak in social studies and

humanities.")

Using what you have learned in this course, how probable should you think it is that Bill is an

accountant? Justify that answer. Do you think most people would agree with that answer?

Answer: The probability that Bill is an accountant should be a slightly greater than 25% percent. The description of Bill leads one to think that he is an accountant, but because 75% of the files in the drawer belonged to sales people, one should not say that the probability of Bill being an accountant is therefore much greater than 25%. Most people probably would not agree with this answer, and would say that the probability of Bill being an accountant should be much more probable (say 75% or so). This is the result of giving greater weight to the description of Bill, and ignoring the fact that 75% of the applicants were sales people, and only 25% were accountants.

4. Bjorn Borg

This problem comes from an experiment conducted when Bjorn Borg was winning

everything.

"Suppose Bjorn Borg reaches the Wimbledon finals in 1981. Please rank order the following

outcomes from most to least likely:

Borg will win the match - 1

Borg will lose the first set - 2

Borg will win the first set but lose the match - 4

Borg will lose the first set but win the match" - 3

First, provide your own rank orderings to the above (even though you don't know much

about Borg). Secondly, discuss what the cognitive psychologists were probably looking for

when they designed the above question and what they most likely found.

Answer: Since Borg was winning everything at this time, the probability of him winning the match is the most likely. The probability of him losing the first set is the next most probable event, since it is a single event, and you can lose the first set and still win the match. Since “Borg will lose the first set but win the match” contains the first and second most probable events, this is ranked third, and finally, since “Borg will win the first set but lose the match” contains the least probable events, this is ranked fourth.

When cognitive psychologists designed the above question, they were probably looking to see how people ranked the probability of single events compared with compound events. One likely result would be that people ranked “Borg will win the match” as most likely and “Borg will lose the first set” as least likely, but then ranked “Borg will lose the first set but win the match” as either the second or third most likely alternative, which contradicts the rules of probability.

5. The Hit and Run Cab

A cab was involved in a hit and run accident at night. Two cab companies, the Green and

the Blue, operate in the city but Green has the bigger fleet, nine times bigger to be precise.

a) At first the police can find no witness. At this point what is a reasonable value to assign to

the probability that the hit and run cab was Green? Explain briefly.

Since we don’t have any additional information, we must rely on the number of green and blue cabs in the city, and say that the probability of the cab being green is 90%.

b) The police now find a witness, but he is badly color blind and hasn't the foggiest idea

what color the cab was. "But it was going too fast," he says. At this point, what is a

reasonable probability to assign to the hypothesis that the cab was Green?

Explain briefly.

90% - for the same reason as in A.

c) Another witness comes forward now and testifies that the cab was Blue. Lawyers for the

cab company try to impugn her testimony by testing the reliability of the witness under the

circumstances that existed on the night of the accident and concluded that the witness

correctly identified each of the colors 80% of the time and failed 20% of the time.

Which, if any, of the following statements correctly summarizes part or all of the findings of

the reliability test:

I) prob of reporting green, given that the cab was in fact green, is 80%

This is true. When we know the color of the cab, we know that her report is correct 80% of the time.

2) prob that the cab was in fact green, given that the witness reported it as such, is 80%

This is false (this is the inverse of Statement #1). We can predict the accuracy of the report if we know the color of the cab, but we can’t predict the color of the cab from her report. The probability that the cab is green, given that the witness reported it as such, is not 80%. In determining the probability that the cab was green, we also have to take into the consideration the fact that 90% of the cabs in the city are green.

3) you can believe what the witness says 80% of the time - if she says the cab was Blue,

you can bet that it was blue and be right four times out of five.

This is also false, for the same reason as in #2.

4) 80% of the time, the witness reports the correct color - if the cab is a Blue one, four to

one she'll call it Blue.

This is true, for the same reason as in #1.

5) If you put the witness out on a street corner on a night like that and had her watch the

cabs go by, 80% of the time she would report green and 20% of the time she would report

blue - on average.

This is false since 90% of the cabs are green and she is equally accurate in reporting both colors.

Extra Credit:

This is a standard Bayes’ Theorem problem. The base rate for Green is 90% but the Report that the cab was blue pulls in the opposite direction.

P (G/R) = ____p (G) x p(R, G)______________

[p (G) x p(R, G)] + [p(B) x p(R,B)]

= ____ 0.9 x 0.2 ______________

[ 0.9 x 0.2] + [ 0.1 x 0.8]

= 9/13

The answer still favors Green but by much less than 90%.