Question

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 $\alpha$ be the probability that the tumor is cancerous; let $\beta$ be the conditional probability that the tumor is cancerous given that the doctor does not call. (a) Which should be larger, $\alpha$ or $\beta ?$ (b) Find $\beta$ in terms of $\alpha,$ and prove your answer in part (a). 3.32. A family has $j$ children with probability $p_{j},$ where $p_{1}=.1, p_{2}=.25, p_{3}=.35, p_{4}=.3 .$ A child from this fam- ily is randomly chosen. Given that this child is the eldest child in the family, find the conditional probability that the family has (a) only 1 child; (b) 4 children. Redo (a) and (b) when the randomly selected child is the youngest child of the family.

Name: Problem 31, Chapter 3: A First Course in Probability
Uploaded: 2021-08-05T18:50:25-08:00
Duration: 5 min 28 s
Channel: Carolyn Behr-Jerome

   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 $\alpha$ be the probability that the tumor is cancerous; let $\beta$ be the conditional probability that the tumor is cancerous given that the doctor does not call.
(a) Which should be larger, $\alpha$ or $\beta ?$
(b) Find $\beta$ in terms of $\alpha,$ and prove your answer in part (a).
3.32. A family has $j$ children with probability $p_{j},$ where $p_{1}=.1, p_{2}=.25, p_{3}=.35, p_{4}=.3 .$ A child from this fam-
ily is randomly chosen. Given that this child is the eldest child in the family, find the conditional probability that the family has
(a) only 1 child;
(b) 4 children.
Redo (a) and (b) when the randomly selected child is the youngest child of the family.

A First Course in Probability

Sheldon Ross 9th Edition

Chapter 3, Problem 31

Instant Answer

Solved by Expert Carolyn Behr-Jerome

08/05/2021

Step 1

Aquina's situation. Ms. Aquina wants to avoid bad news over the weekend. She instructs the doctor to flip a coin. If heads, the doctor calls if the news is good and does not call if the news is bad. If tails, the doctor does not call regardless of the news. Show more…

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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 $\alpha$ be the probability that the tumor is cancerous; let $\beta$ be the conditional probability that the tumor is cancerous given that the doctor does not call. (a) Which should be larger, $\alpha$ or $\beta ?$ (b) Find $\beta$ in terms of $\alpha,$ and prove your answer in part (a). 3.32. A family has $j$ children with probability $p_{j},$ where $p_{1}=.1, p_{2}=.25, p_{3}=.35, p_{4}=.3 .$ A child from this fam- ily is randomly chosen. Given that this child is the eldest child in the family, find the conditional probability that the family has (a) only 1 child; (b) 4 children. Redo (a) and (b) when the randomly selected child is the youngest child of the family.

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Key Concepts

Discrete Probability Distributions

Discrete probability distributions describe the probabilities of outcomes in a process that has a countable number of possible results. They are key to analyzing scenarios such as the varying sizes of families and the corresponding probabilities attached to having a specific number of children, especially under conditions where the order (eldest or youngest) influences the probability calculations.

Randomized Response Mechanism

The randomized response mechanism is a method used to introduce randomness into the communication of sensitive information. In the context of the problem, it is used to obscure the direct link between the actual test result and the doctor’s action by using a coin flip, thereby altering the inferential power of a non-call message.

Law of Total Probability

The Law of Total Probability allows one to compute the overall probability of an event by partitioning the sample space into several mutually exclusive events. This is crucial when considering multiple scenarios (e.g., different coin outcomes or child ordering scenarios) that contribute to the occurrence of the observed event.

Conditional Probability

Conditional probability is the measure of the likelihood of an event occurring given that another event has already occurred. It is essential for understanding how additional information (such as not receiving a call from the doctor, or the selected child being the eldest) updates the probability of an underlying event (the tumor being cancerous or the family having a certain number of children).

Bayes' Theorem

Bayes' Theorem provides a formal mechanism for updating probabilities based on new evidence. It relates the conditional and marginal probabilities of events, and it is especially useful for problems where prior beliefs (such as the likelihood of cancer) are revised after observing outcomes (like a doctor’s call or lack thereof).

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