Download Answers to In-Class Activity #7

LSP 121-405
In-Class Exercise #7
Risks and Test Errors
20 points
Due by Thursday, Oct. 29th at 11:50 AM
ANSWERS
1. In 1953, French economist Maurice Allais studied how people assess risk and expected
values. He asked a large group of people to select either Option A or Option B for each
of the two survey questions below:
Survey Question 1
Option A: 100% chance of gaining $1,000,000
Option B: 10% chance of gaining $2,500,000; 89% chance of gaining $1,000,000;
and 1% chance of gaining nothing
Survey Question 2
Option A: 11% chance of gaining $1,000,000 and 89% chance of gaining nothing
Option B: 10% chance of gaining $2,500,000 and 90% chance of gaining nothing
Allais discovered that for Survey Question 1, most people chose option A, while for
Survey Question 2, most people chose option B.
a. For each Survey Question, find the expected value of each option.
For each option, the expected value (also called the average gain) is the sum
of the <outcome probability> x <outcome value> for all possible outcomes.
For Survey Question 1, Option A – there is 1 possible outcome and the
expected value is (100% * $1,000,000) = $1,000,000
For Survey Question 1, Option B – there are 2 possible outcomes and the
expected value is (10% * $2,500,000) + (89% * $1,000,000) + (1% * $0) =
$1,140,000
For Survey Question 2, Option A – there are 2 possible outcomes and the
expected value is (11% * $1,000,000) + (89% * $0) = $110,000.
For Survey Question 2, Option B – there are 2 possible outcomes and the
expected value is (10% * $2,500,000) + (90% * $0) = $250,000
b. Are the responses given in the surveys consistent with the expected values?
For Survey Question 1, the answers are not consistent with the expected
values because the expected value of Option B is higher than the expected
value of Option A, yet most people chose Option A.
For Survey Question 2, the answers are consistent with the expected values
because the expected value of Option B is higher than the expected value of
Option A, and most people chose Option B.
c. Give a possible explanation for the responses in Allais’ surveys.
Many people would prefer to choose a “sure thing”, such as option A in
Question #1, where there is no possibility of losing. They choose Option A
because it has no risk, even though Option B has a higher expected outcome.
2. The numbers for a particular type of cancer are as follows:
1 in 1000 tumors are malignant
A blood test for determining malignancy is 90% accurate and is given to 15,000
people with this kind of tumor
Create a matrix similar to the one in the notes which displays true positives, false
positives, false negatives, true negatives, and all totals. Paste this matrix into your Word
document.
First, we create the table. On the bottom right corner we put the total number of
tests – 15,000. Then we fill in the bottom row – since we know that 1 in 1000 tumors
are malignant, the total number of malignant tumors is (1/1000) * 15,000 = 15. The
rest of the 15,000 tumors are benign = 15,000 – 15 = 14,985.
Tumor is Tumor is
Malignant Benign
Total
Test says it
is Malignant
Test says it
is Benign
Total
15
14,985
15,000
For each column, we multiply the total at the bottom by 90% and this gives us the
number of tests that were correct (true positives in first column, true negatives in
second column).
Tumor is
Malignant
Test says it
is Malignant
Total
13.5
(true
positives)
13486.5
Test says it
is Benign
Total
Tumor is
Benign
(true
negatives)
15
14,985
15,000
Now, for each column, we subtract from the total at the bottom to fill in the false
positives and false negatives and add the rows to fill in the totals at the end.
Test says it
is Malignant
Test says it
is Benign
Total
Tumor is
Malignant
Tumor is
Benign
13.5
1498.5
(true
(false
positives) negatives)
1.5
Total
1,512.0
13486.5
13,488.0
(false
(true
positives) negatives)
15
14,985
15,000
a. What is the chance that a positive blood test really means your patient has cancer?
There are 1512 positive blood tests (indicating malignancy) and only 13.5 of
these actually have cancer, so the chance that a positive blood test really
means that the patient has cancer is 13.5 / 1512 = 0.009 = 0.9%
b. Assume the blood test results were negative for your patient. What is the chance
that your patient actually has cancer?
There are 13,488 negative blood tests (indicating no malignancy) and 1.5 of
these actually have cancer, so the chance that a negative blood test really
means that the patient has cancer is 1.5 / 13488 = 0.00011 = 0.011%
.