3. Solve below: A. A new test to detect TB has been designed. It is estimated th

Question

3. Solve below:

A.

A new test to detect TB has been designed. It is estimated that 88% of people taking this test have the disease. The test detects the disease in 97% of those who have the disease. The test does not detect the disease in 99% of those who do not have the disease. If a person taking the test is chosen at random, what is the probability of the test indicating that the person does not have the disease?

a) 0.1452

b) 0.9900

c) 0.0100

d) 0.0300

e) 0.0264

f) None of the above.

B.

5 yellow balls and 7 red balls are placed in an urn. Two balls are then drawn in succession without replacement. What is the probability that the first ball drawn is a red ball if the second ball drawn is yellow?

a) 0.2449

b) 0.6667

c) 0.9091

d) 0.4167

e) 0.6364

f) None of the above.

C.

There are three colored cookie jars. One jar is blue, another green and the last one pink. The blue jar contains 10 chocolate chip and 7 sugar cookies. The green jar contains 8 chocolate chip, 14 sugar and 11 peanut butter cookies. The pink jar contains 6 chocolate chip, 5 sugar and 9 peanut butter cookies. One of the three cookie jars is chosen at random. The probabilities that the blue jar, green jar, or pink jar will be chosen are 12 , 14 , and 14 , respectively. A cookie is then chosen at random from the chosen jar. What is the probability that the pink jar was chosen, if it is known that the cookie was a sugar cookie?

a) 0.0833

b) 0.2302

c) 0.0625

d) 0.1669

e) 0.2500

f) None of the above.

D.

An experiment consists of choosing an urn with the following probabilities that Urn 1, Urn 2, or Urn 3 will be chosen: 1/2, 1/4, and 1/4, respectively. Urn 1 contains 15 brown marbles and 5 clear marbles. Urn 2 contains 12 brown marbles, 9 clear marbles and 14 red marbles. Urn 3 contains 7 brown marbles, 6 clear marbles and 8 red marbles.   

A marble is then chosen from the chosen urn. What is the probability that Urn 3 was chosen, given that the marble chosen was clear?

a) 0.2857

b) 0.0952

c) 0.2740

d) 0.3604

e) 0.0714

f) None of the above.

Explanation / Answer

a) 88% are carriers, so 12% are not.
Of the 88%, 97% test positive; that's (.88)(.97) = 0.8536 = 85.36% of the testing pool who give true positives.
The other (.88)(.03) = .0264 = 2.64% of the testing pool give false negatives.

Of the healthy 12%, 99% give true negatives; that's (.12)(.99) = 0.1188 = 11.88% of the testing pool.
The remaining (.12)(.02) = .0024 = 0.24% of the testing pool give false positives.
The people in whom the test shows no disease are the true negatives (11.88%) and the false negatives (2.64%); so, a total of 14.52% of the testing pool test clear.

Answer is 0.1452

B) OPTION (e)

P[1st red & 2nd yellow] = 7/12 * 5/11 = 0.2651 = A
P[1st yellow & 2nd yellow] = 5/12 * 4/11 = 0.1515
P[2nd yellow] = sum of the above = 0.4166 = B
P[1st red | 2nd yellow] = A/B = 0.2651/0.4166= 0.6364

c) P[ pink jar & sugar cookie] = 1/4*5/20 = 0.0625
P[sugar cookie] = 1/2*7/17 + 1/4*14/33 + 1/4*5/20 = 0.6097
P[pink | sugar cookie] = 0.0625/0.6097= 0.1025

OPTION F (NONE OF THESE)

D) P(clear) = sum of P(urn @) * P(clear | urn @)

P(clear) = 1/2 (5 / 20) + 1/4 (9 / 35) + 1/4 (6 / 21)
P(clear) =5 / 40 + 9/ 140 + 6 / 84
P(clear) = 0.26
of this total probability of a clear marble, 1/4 (6/21) = 6/84 = 0.0714 comes from urn 3

so P(urn 3 | clear) =0.0714/0.26 = 0.2740

OPTION C

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