college algebra

COLLEGE ALGEBRA QUESTIONS ON PROBABILITY. aNY ONE WILLING TO HELP OUT/

 

WHAT

IS PROBABILITY?

Insight into the use of probability in the medical community:

Probability is a recurring theme in medical practice. No doctor who returns home from a busy day at the hospital is spared the nagging feeling that some of his diagnoses may turn out to be wrong, or some of his treatments may not lead to the expected cure. Encountering the unexpected is an occupational hazard in clinical practice. Doctors after some experience in their profession reconcile to the fact that diagnosis and prognosis always have varying degrees of uncertainty and at best can be stated as probable in a particular case.

Critical appraisal of medical journals also leads to the same gut feeling. One is bombarded with new research results, but experience dictates that well-established facts of today may be refuted in some other scientific publication in the following weeks or months. When a practicing clinician reads that some new treatment is superior to the conventional one, he will assess the evidence critically, and at best he will conclude that probably it is true.

Types of probabilities

Suppose that we want to determine the probability of obtaining an ace from a pack of cards (which, let us assume has been tampered with by a dishonest gambler), we proceed by drawing a card from the pack a large number of times, as we know in the long run, the observed frequency will approach the true probability. Since there are 4 aces in a 52 card deck, the real probably of drawing an ace on a single draw would be ¼ or 7.69%. Mathematicians often state that a probability is a long-run frequency. Consider the statement, “The cure for Alzheimer’s disease will probably be discovered in the coming decade.” This statement does not indicate the basis of this expectation or belief. However, it may be based on the present state of research in Alzheimer’s. A probabilistic statement incorporates some amount of uncertainty, which may be quantified as follows: A politician may state that there is a fifty-fifty chance of winning the next election.

Consider a 26-year-old married female patient who suffered from severe abdominal pain is referred to a hospital. She is also having amenorrhea for the past 4 months. The pain is located in the left lower abdomen. The gynecologist who examines her concludes that there is a 30% probability that the patient is suffering from ectopic pregnancy.

If you were to ask the gynecologist to explain on what basis the diagnosis of ectopic pregnancy is suspected, the Dr. might state that he/she has studied a large number of successive patients with this symptom complex of lower abdominal pain with amenorrhea, and that a subsequent laparotomy revealed an ectopic pregnancy in 30% of the cases.

If we accept that the study cited is large enough to make us assume that the possibility of the observed frequency of ectopic pregnancy, it is natural to conclude that the gynecologist’s probability claim is ‘evidence based’.

Medical False Positives and False Negatives

Summary: If a test for a disease is 99% accurate, and you test positive, the probability you actually have the disease is not 99%. In fact, the more rare the disease, the lower the probability that a positive result means you actually have it, despite that 99% accuracy. The difference lies in the rules of conditional or contingent probability.

You’ve taken a test for a deadly disease D, and the doctor tells you that you’ve tested positive. How bad is the news? You need to know how accurate the test is, and specifically you need to know the probability that your positive test result actually means you have the disease.

Suppose you’re told the test for D is “99% accurate” in the following sense: If you have D, the test will be positive 99% of the time, and if you don’t have it, the test will be negative 99% of the time. The other 1% in each scenario would be a false negative or false positive. Suppose further that 0.1% — one out of every thousand people — have this rare disease.

You might think that a positive result means you’re 99% likely to have the disease. But 99% is the probability that if you have the disease then you test positive, not the probability that if you test positive then you have the disease.

Additional Insight:

Any medical diagnosis does not arrive in an instant. Instead, physicians use a step-by-step process that is influenced by an understanding of probability.

An experienced medical practitioner begins the process of making a diagnosis upon first laying eyes on a patient, and probability is one of the main tools they use in this process.

The diagnostic process can begin even before laying eyes on the patient. Suppose that you were asked to diagnose a patient that you have not seen yet and is still in the waiting room. Sounds crazy? Consider the fact that you may already know a lot about the patient, without even seeing them and can make some educated guesses. You could realistically already know that the patient is a 48-year-old woman referred by a family doctor because of migraine headaches.

You might consider your experience with other women in their forties that you have seen with the same ailment. You might already know, based on your experience and research that about a 25% of patients that you see have a migraine, another third have medication-overuse headaches, and the remainder fall into an “everything else” category that includes tension-type headaches, arthritis of the neck or jaw-joints, sinus disease, tumors, etc. So before seeing the patient I’m already able to identify the two most likely diagnoses and assign an initial probability for each. Make sense?

As a result, you can apply an initial probability to this patient without even seeing the patient. During the actual patient exam to follow, and supplemental testing the initial probability will undergo a series of upward and downward adjustments according to what the patient has to say and what does or does not turn up on her physical examination and testing.

So the next steps in the patient diagnosis process are that the patient comes into the examining room and you listen to her story. In the headache example given, one key piece of data is how many days per month she takes an as-needed medication — for example, aspirin, acetaminophen or a prescription drug. If she takes as-needed medicine more days than not and has been doing so for a matter of months, then the initial probability that you established as to a cause for her ailment, gets adjusted upward or downward. It now becomes a matter of gathering data refine the diagnosis.

In the study of probability, there is a technique that uses concepts inherent in what is known as Bayes’ theorem. Bayes’ theorem states that the probability of a diagnosis after a new fact is added depends on what its probability was before the new fact was added. When a diagnosis is not 100% likely at the time of initial evaluation, the patient’s course of symptoms over time provides yet another form of data that can lead to revision of diagnostic probabilities.

Consider the following scenario:

A patient is admitted to the hospital and a potentially life-saving drug is administered. The following dialog takes place between the nurse and a concerned relative.

RELATIVE: Nurse, what is the probability that the drug will work?

NURSE: I hope it works, we’ll know tomorrow.

RELATIVE: Yes, but what is the probability that it will?

NURSE: Each case is different, we have to wait.

RELATIVE: But let’s see, out of a hundred patients that are treated under similar conditions, how many times would you expect it to work?

NURSE(somewhat annoyed): I told you, every person is different, for some it works, for some it doesn’t.

RELATIVE(insisting): Then tell me, if you had to bet whether it will work or not, which side of the bet would you take?

NURSE(cheering up for a moment): I’d bet it will work.

RELATIVE(somewhat relieved): OK, now, would you be willing to lose two dollars if it doesn’t work, and gain one dollar if it does?

NURSE(exasperated): What a sick thought! You are wasting my time!

Primary Task

For the DB assignment this week, please respond to the following questions:

a) In relation to my presentation above “insight into the use of probability in the medical community”, comment on some of the key points that were discussed and what might have been of interest to you or ideas presented that prompted further thinking on your part.(200 word minimum response is expected).

b) What is your impression of this dialogue between the relative and the nurse? Explain your response in detail. Did you find it amusing? Do you think that kind of conversation could actually occur?

c) Taking on the nurse’s role from the above dialogue, would you have responded in a different manner based on what you know about probability concepts? Explain your thoughts in detail.

d) Based on my presentation above in relation to insight into the use of probability in the medical community, do you believe that the nurse ended the dialogue appropriately? Why?

e) If you would have ended the dialogue in a different manner, provide an amusing and informative ending to the dialogue that you believe would have satisfied the relative’s concerns. We all need a good laugh as this course comes to a close! I may even offer some extra points if your response makes me laugh.