Suppose that 120 people are tested for some disease D. Below is a table representing the outcome of these
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1. Disease Test: Suppose that 120 people are tested for some disease D. Below is a table representing the outcome of these tests. Let T denote the outcome that the test is positive (implying the disease is present) and let D denote the outcome that the disease actually is present.
Disease (D)  No Disease  Total  
Positive Test (T)  17  5  22 
Negative Test  1  97  98 
Total  18  102  120 
For example, the table shows that 17 individuals have the disease and were tested positive. So:
P(D ∩ T) = = 0.142 = 14.2%
Note that ∩ can be interpreted as “and” and ∪ can be interpreted as “or”.
Using the table above and the rules of probability, address the following:
a) Find the probability that an individual has the disease, P(D).
b) Find the probability that an individual is tested positive, P(T^{C}).
c) Find the probability that an individual has the disease and tested negative. Be sure to use set notation.
d) Find the probability that an individual has the disease or tested positive. Use set notation.
e) Another way to calculate P(D ∪ T) in part (d) is to use the complement of the event. First, in words what is the complement to the event? Use this to then find the probability of the event. Does this match your answer in (d)?
f) Suppose that an individual tested positive for the disease. What is the probability that he actually has the disease? (Hint: this is a conditional probability. You can use the definition of conditional probability, or you can think of it the following way: Restrict yourself to those that tested positive and think, what fraction of only those individuals actually have the disease).
g) In diseasetesting situations, there are many important values for researchers to understand. The following are a few of those:
False positive: the event that a test is positive, given that there is in fact no disease present
True positive: the event that a test is positive, given that the disease is present
False negative: the event that a test is negative, given that the disease actually is present
True negative: the event that the test is negative, given that there is no disease present
Using this information, translate the above events into set notation using the symbols D and P, and then find the probability of those events. The first one has been done for you as an example.
Event in words  Event in set notation  Probability 
False positive  P(TD^{C})  P(TD^{C}) = = 0.049 
True positive  P(TD)  P(TD) = 17/18 = 0.944

False negative  P(T^{C}D)  P(T^{C}D) = 1/18 = 0.056

True negative  P(T^{C}D^{C})  P(T^{C}D^{C}) = 97/102 = 0.951

h) Given the current scenario, which do you think is worse: a false positive or a false negative? Why?
2. Lottery Suppose that you enter two small lotteries, A and B, to win a new statistics textbook. You have been told that you have a 15% chance of winning in lottery A and a 40% chance of winning in lottery B. You also know that your chance of winning both is 6%.
a) Are the events {winning A} and {winning B} independent? Why or why not? Does this finding seem reasonable?
b) Are the events {winning A} and {winning B} mutually exclusive? Why or why not?
c) Without doing any calculations, determine the probability of winning A given that you’ve won in lottery B. How did you know this?
d) Find the probability that you will win either lottery A or B (Note: this is also the probability that you will win at least one textbook).
e) Find the probability that you will not win a textbook. What is the relationship of this event with the one in part (d)? (Hint: find the relationship first.)
 9 years ago
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