4 question in OR/ Probability

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nmzrfhgna

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please solve these 4 questions in Operation Research/Probability

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4Of all 40-year –old women. 1% have breast cancer. If a woman has breast cancer. a mammogram will give a positive indication for cancer 90% of the time. if a women does not have beast cancer, a mammogram will give a positive indication for cancer 9% of the time . if a 40- year –old women’s mammogram gives a positive indication for cancer, what is the probability that she has cancer? 5three out of every 1,000 low – risk 50 –old males have colon, if a man has colon cancer a test for hidden blood in the stool will indicate hidden blood half the time if he does not have colon cancer . a test for hidden blood in the stool will indicate hidden blood 3% of the time if the hidden – blood lest turns out positive for a low – risk 50- year -old male what is the chance that he has colon cancer 6you have made it to the final round of “let’s Make a Deal “ you know there is SI million behind cither door 1, door2, door 3, It is equally likely that the prize is behind any of the three the tow doors without a prize have nothing behind them , you randomly choose door 2, but before door 2 is opened Monte reveals that there is no prize behind door 3. You now have the opportunity to switch and choose door1. Should you switch? Assume that Monte plays as empty door but he cannot open door 2. Monte is equally behind door 1. Or door 3. If the prize is really behind door 1Monte open door 3. If the prize is really behind door 3. Monte must open door 1. What is your decision 3A customer has approached a bank for a loan Without further information, the bank believes there is a 4% chance that the customer will default on the loan. The bank can run a credit check on the customer. the check will yield either a favorable or an unfavorable report. From the past experience, the bank believes that P(favorable report being received )customer will default=1/40, and P( favorable report customer will default )=99/100. If a favorable report is received, what is the probability that the customer will default on the loan.
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Explanation & Answer

Hi azmesutan, Please check the attached file for details, let me know if you have any question. Thank you, BestJames,

4Of all 40-year –old women. 1% have breast cancer. If a woman has breast cancer, a
mammogram will give a positive indication for cancer 90% of the time. If a women does not
have beast cancer, a mammogram will give a positive indication for cancer 9% of the time . if a
40-year -old women’s mammogram gives a positive indication for cancer, what is the
probability that she has cancer?
Soution:
Let p be a 40-year-old woman has positive in mammogram, and c be a 40-year-old woman has
breast cancer, then C~ be a 40-year-old woman does not have breast cancer. Then we have P(c)
= 0.01, P(c~) = 0.99, P(p|c) = 0.90, P(p|c~) = 0.09 , we need to find P(c|p). Using Bayes
theorem, we have
P(c|p) = P(p|c)*P(c)/P(p) = P(p|c)*P(c)/(P(p|c)*P(c)+ P(p|c~)*P(c~))
=0.90*0.01/(0.90*0.01+0.09*0.99)
=0.0917

5three out of every 1,000 low – risk 50 –old males have colon, if a man has colon cancer a test
for hidden blood in the stool will indicate hidden blood half the time. If he does not have colon
cancer, a test for hidden blood in the stool will indicate hidden blood 3% of the time. If the
hidden – blood lest turns out positive for a low – risk 50- year -old male, what is the chance that
he has colon cancer
Solution:
Let c be a low-risk 50-old male have colon cancer, then P(c) = 0.003, P(c~) = 0.997. let b be the
event that test indicate the hidden blood in the stool, then P(b|c) =0.5, P(b|c~) = 0.03. We
need to find P(c|b). Using Bayes theorem,
P(c|b) = P(b|c)P(c)/P(b) = P(b|c)*p(c...