# Probability

(right or not )

1. A student takes a multiple choice exam.suppose for each questions he either knows the answer or gambles and chooses an option at random. further suppose that if he knows the answer, the probability of a correct answer is 1, and if gambles the probability is 1/4. to pass, the student needs to answer at least 60% of the questions correctly.

the student has 'studied for a minimal pass,' that is, with probability 0.6 he knows the question. Given that he answers a question correctly, what is the probability that he actually knows the answer? (20/23 ?)

2. the probability of getting a rare of cancer is 0.01. a clinical test can be used to test whether a person has the cancer. if a person has the cancer, the probability that test positive is 0.98. if a person does not have the cancer, the probability that test negative is 0.95.

a) given that a person is tested positive, what is the probability he actually has the cancer?(98/593 ?)

b) if a person is tested positive, and again with another positive result, what is the probability he actually has the cancer?

### 2 個解答

• 最愛解答

1)P(he actually knows the answer | he answers a question correctly)= P(he actually knows the answer | he actually knows the answer or gambles successfully)= 0.6(1) / [0.6(1) + (1 - 0.6)(1/4)]= 6 / 7 (I think the answer is not 20/23)

2a) P(he actually has the cancer | he is tested positive)= P(he actually has the cancer and is tested positive | he actually has the cancer and is tested positive OR he does not have the cancer and is tested positive)= (0.01) 0.98 / [(0.01) 0.98 + (1 - 0.01) (1 - 0.95)]= 98 / (98 + 99 * 5)= 98 / 593

b) P(he actually has the cancer | he is tested positive twice)

= P(he actually has the cancer and is tested positive twice | he actually has the cancer and is tested positive twice OR he does not have the cancer and is tested positive twice)

.................. (0.01) 0.98²

= ──────────────────────

....(0.01) 0.98² + (1 - 0.01) (1 - 0.95)²

..............98²

= ──────────

......98² + 99 * 5²

= 9604 / 12079

• (◕‿◕) 我今次最快都只是排第二咋～ (=^o^=)

Q2 (b) 是好的題型，可以好好參考再作日後修改。