Situation: The High School and Beyond data is from a large-scale longitudinal study conducted by the National Opinion Research Center (1980) under contract with the National Center for Education Statistics. Below is a table representing a sample of 100 students from this data that includes the student's gender and whether the high school they attended was public or private. Let A denote the subset of these 100 subjects that the student is Female, so the event A = {Female}. Similarly define event B as the student attended a Public high school, so the event B = {Public}.. The following 2x2 table classifies each student according to gender and school type.
|
|
Public
|
Private
|
Total
|
|
Female
|
38
|
7
|
|
|
Male
|
46
|
9
|
|
|
Total
|
|
|
100
|
For example, the table reveals that 38 students were both Female and attended a Public high school, so:
P(A ∩ B) = 38/100 = 0.38 (read "the probability of A and B is 0.38)).
a. Fill in the marginal totals of the table (the row and column totals). From these totals determine the probability that the student is Female and also the probability the student attended a Public high school. Answer the following placing the proper event notation (e.g. A, B) in the ( ).
P(Female) = P( ) =
P(Public) = P( ) =
b. Translate the following events into set notation using the symbols A and B, complement, union, intersection. Also give the probability of the event as determined from the table above. Fill in the table below with these values. {If you cannot construct the symbol ∩, just use & or copy and paste} Lastly, draw a Venn diagram (on scratch paper, but you need not submit it) showing the events of Female, Public, and both Female and Public. The first one is completed for you.
|
Event in words
|
Event in set notation
|
Probability
|
|
Female and Public
|
A ∩ B
|
P(A ∩ B) = 0.38
|
|
Female and Private
|
|
|
|
Public and Male
|
|
|
|
Neither Female nor Public
|
|
|
c. Determine the probability that the student is Male. Do the same for student attending a Private high school. Use proper event notation remember that A is Female and B is Public.
d. If you had not been given the table, but instead had merely been told that P(A)=.45 and P(B)=.84, would you have been able to calculate P(Ac) and P(Bc)? Explain how.
e. Calculate the probability that at least either the student is Female or the student attended a Public high school.
f. Another way to calculate P(A ∪ B) in part e is to use the complement of the event. First, in words what is the complement to the event described in e and then find this probability using the complement. Does this match your answer in e?
g. If you had not been given the table but instead had merely been told P(A)=.45 and P(B)=.84 and were asked to calculate P(A ∪ B), you might first consider P(A)+P(B). Calculate this sum and compare the result to your answer for e. Are they the same? Is this even a legitimate answer?
h. Given the knowledge that the student is Female, what is the conditional probability that the student attended a Public high school? [Hint: Restrict your consideration to Females, and ask yourself what fraction of those female students attended a Public high school?]
i. How does this conditional probability of the student attending a Public high school given that the student is Female compare with the unconditional probability of the student attending a Public high school? Does the knowledge that the student is Female make it more or less likely (or neither) that the student attended a Public high school?
1.2 Graduate School Admissions Suppose that you apply to two graduate schools A and B, and that you believe your probability of acceptance by A to be 0.7, your probability of acceptance by B to be 0.6, and your probability of acceptance by both to be 0.5.
a. Are the events {acceptance by A} and {acceptance by B} independent? Explain. {Hint: look at the definition for independence from the lecture notes or in the Probability Rules table in the chapter.}
b. Determine the conditional probability of acceptance by B given acceptance by A? How does it compare to the (unconditional) probability of acceptance by B?
c. What is the probability that you are accepted by at least one of the two schools?
1.3 Just for fun! Back in the Seventies there was a game show called "Let's Make a Deal". The rules were simple: a contestant was selected from the crowd and offered the chance to open one of 3 doors. Behind one door was a prize (e.g. money, new car), while behind the other two doors were gag prizes (e.g. a donkey, barrel of popcorn). After making their choice of door the game show host would open one of the non-selected doors which was always one that did not contain the prize. The contestant was then asked if he or she would like to switch from their first choice to the remaining unopened door or stay with their original choice. If you were the contestant, what choice would you make? That is, given that you now have two doors unopened, one with a prize behind it and the other without, which gives you a better chance of winning: switching doors or staying with your original choice? After you have thought about this, please visit the following website and try it out! {You may have to click the refresh button - after opening the page}