Solution Found!
Ms. Aquina has just had a biopsy on a possibly cancerous
Chapter 3, Problem 31P(choose chapter or problem)
Problem 31P
Ms. Aquina has just had a biopsy on a possibly cancerous tumor. Not wanting to spoil a weekend family event, she does not want to hear any bad news in the next few days. But if she tells the doctor to call only if the news is good, then if the doctor does not call, Ms. Aquina can conclude that the news is bad. So, being a student of probability, Ms. Aquina instructs the doctor to flip a coin. If it comes up heads, the doctor is to call if the news is good and not call if the news is bad. if the coin comes up tails, the doctor is not to call. In this way, even if the doctor doesn’t call, the news is not necessarily bad. Let α be the probability that the tumor is cancerous; let β be the conditional probability that the tumor is cancerous given that the doctor does not call.
(a) Which should be larger, α or β?
(b) Find β in terms of α, and prove your answer in part (a).
Questions & Answers
QUESTION:
Problem 31P
Ms. Aquina has just had a biopsy on a possibly cancerous tumor. Not wanting to spoil a weekend family event, she does not want to hear any bad news in the next few days. But if she tells the doctor to call only if the news is good, then if the doctor does not call, Ms. Aquina can conclude that the news is bad. So, being a student of probability, Ms. Aquina instructs the doctor to flip a coin. If it comes up heads, the doctor is to call if the news is good and not call if the news is bad. if the coin comes up tails, the doctor is not to call. In this way, even if the doctor doesn’t call, the news is not necessarily bad. Let α be the probability that the tumor is cancerous; let β be the conditional probability that the tumor is cancerous given that the doctor does not call.
(a) Which should be larger, α or β?
(b) Find β in terms of α, and prove your answer in part (a).
ANSWER:
Step 1 of 2
(a)
We are asked to find which should be a larger, or
Let denote the event that Ms. Aquina has cancer and the doctor has bad news.
Let be the event that Ms. Aquina does not have cancer and the results of the test are good.
Let be the event that the doctor calls the house during the holiday.
Now the event that the doctor does not call (i.e. ) will say that Ms. Aquina has cancer or event
if and only if it is more likely that the doctor won’t call given that the Ms. Aquina does have cancer.
This is the event will cause
to be greater than
if and only if
Hence all possible outcomes we have that
Since if Ms. Aquina has cancer, the doctor will not call regardless of the coin flip according to the question.
Hence we can write,
Since if Ms. Aquina does not have cancer, the doctor will only call if the coin flip lands heads and not call otherwise.
Thus the fact that the doctor does not call prove that Ms. Aquina has cancer.
Hence is larger than