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.

Question

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.
A ∩ B
.
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?

Answer 1

this is the chart for this problem
Public Private Total
Female 38 7 45
Male 46 9 55
Total 84 16 100

Answer 2

P(B|A) = (P(A and B))/(P(A)) = (.38)/(.45) = 0.84