Ms. Aquina has just had a biopsy on a possibly can- cerous 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 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 flop 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 to (a).

a) Refer to the diagram attached to the bottom of this answer. Given that the tumor is cancerous (that is saying that α happens), there are other events different than β in that event space, so β should be smaller than α.

b) Using the diagram again, we can see that the probability of β happening is the product of α and the probability that the coin shows tails, which is 0.5, then:

β = 0.5·α

which proves β is half of α, hence α is larger.