Probability question: Ms. Aquina has just had a biopsy

[Jamin2112]

Probability question: "Ms. Aquina has just had a biopsy ..."

Homework Statement

Ms. Aquina has just had a biopsy on a possibly cancerous tumor. Not wanting to spoil a weekend family even, 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 if 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 A be the probability that the tumor is cancerous; let B be the conditional probability that the tumor is cancerous given that the doctor does not call. Prove the B > A.

Homework Equations

Baye's formula

The Attempt at a Solution

This has me confused. I'll explain my though process and then you can correct me and perhaps help me prove this for me.

B = P(Tumor is cancerous | Doc doesn't call) = P(Tumor is cancerous & Doc doesn't call) / P(Doc doesn't call) = P(Doc doesn't call | Tumor is cancerous) * P(Tumor is cancerous) / P(Doc doesn't call) = 1 * A / P(Doc doesn't call) = ?

I can't figure out the probability that the Doc doesn't call.

 

[HallsofIvy]

If A is the probability the tumor is cancerous, then 1- A is the probability it is not.

If the tumor is cancerous, the Dr. does not call. If the tumor is not cancerous the probability the Dr. does not call is 1/2. So the probability the Dr. calls is A(0)+ (1- A)(1/2)= 1/2- A/2.

 

[Jamin2112]

That's what I got, but a solution manual online got 3/4.

 

[Jamin2112]

Chapter 3, Problem 31 from this solution manual http://waxworksmath.com/Authors/N_Z/Ross/AFirstCourseInProb/WriteUp/weatherwax_ross_solutions.pdf