Introduction: Different authors use different symbols, terminology, and abbreviations to denote probabilities. For example, we may find:
Probability of Heads, given that it is a Fair Coin, is one-half;
Prob (Heads, Fair Coin) = 1/2;
P(H/F) = 1/2;
P(H) = 1/2 (taking it for granted that the coin is fair);
The frequency of Heads in an indefinitely long series of tosses of a fair coin approaches 50%;
The fraction of Heads in a large sample of coin tosses centers on 1/2.
There are a few basic rules concerning the relationship between probabilities that are referred to in your text and in lecture from time to time. Although you are rarely asked to actually apply them, it may be useful to review them here:
Rule about Negation: P(A) + P(not-A) = 1
Rule about Conjunction: P(A and B) = P(B/A) x P(A)
Rule about Disjunction: P(A or B) = P(A) + P(B) - P(A and B)
In this unit we have used the term "inverse probability". Suppose bar owner is looking at the relationship between ordering a drink (call that D) and nibbling on free chips (call that C). She may note that a large percentage of those with a drink are nibbling on chips. [In other words, P(C/D).] What she may be more interested in, however, is the inverse, P(D/C).
We also say that P(C/D) is the inverse of P(D/C).
By the way, it is likely that what the owner really wants to know are the comparative values of P(D/C) and P(D/not-C). Is the probability of ordering a drink a lot greater when one nibbles chips?
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) = ?
b) P(B) = ?
c) P(O/B) = ?
d) P(B/O) = ?
e) P(not-O) = ?
f) P(not-B) = ?
g) P(not-O/B) = ?
h) P(not-B/O) = ?
i) P(O/not-B) = ?
j) P(B/not-O) = ?
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"
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?
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
Borg will lose the first set
Borg will win the first set but lose the match
Borg will lose the first set but win the match"
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.
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.
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.
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%
2) prob that the cab was in fact green, given that the witness reported it as such, is 80%
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.
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.
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.
Extra Credit: *d) Given all you now know about the hit and run car, what is the probability that it was Green? Show your work.