Day 07, MAT115

• Announcements:
  • Remember: you have an assignment due Friday.

  • You also have a new reading assignment for Friday: Location, Location, Location (p. 35)

• Last time we worked over the reading "Chances Are", in which Strogatz attempts to make conditional probability a little easier. We'll continue with our

  Question of the Day: What happens when you're falsely accused of doping at work?

  Recall that our reading emphasizes is how bad "the experts" really are at probabilities. We rely on them, but maybe we shouldn't when it comes to risk. Let's see if we can do better than the experts!

• Let's recall what we did last time with "Chances Are":

  We started with some vocabulary:

  • conditional probability -- probability that an event happens given that some other thing has already happened
  • "intersection" and Venn Diagram
  • Bayes's Theorem
  Today we'll see if we can't follow Strogatz as he attempts to turn what seems difficult into something reasonable, through "right thinking".

  Last time we worked out "the plant problem":

  • Before going on vacation for a week, you ask your spacy friend to water your ailing plant. Without water, the plant has a 90 percent chance of dying. Even with proper watering, it has a 20 percent chance of dying. And the probability that your friend will forget to water it is 30 percent.
    1. What's the chance that your plant will survive the week?
    2. If it's dead when you return, what's the chance that your friend forgot to water it?
    3. If your friend forgot to water it, what's the chance that it'll be dead when you return?

  • We usually start by defining some things. For example we could let
    • - the plant died
    • - the plant was watered

    then, of course, we have also defined the complements:

    • - the plant didn't die
    • - the plant was not watered

  • We used a Venn diagram to construct the universe.

  • Next we introduced the notion of conditional probabilities: for example,
    The robability of given

    is written as

    

    Here's "Bayes's Theorem", which relates the conditional probabilities to the probability that both of two different things happen in a set:

    

    And because the order of the intersection (both happening) doesn't matter, the problem is symmetric in D and W:

    

    These are useful, provided we know two of the three quantities -- then we can solve for the third.

    And since the two quantities on the left-hand sides are equal, we can equate the right-hand sides:

    

    This is Bayes's Theorem (and is especially useful provided we know three of the four quantities -- then we can solve for the fourth).

    The given probabilities are

    1. 
    2. 
    3. 

  • We can now turn the English questions posed at the beginning of the plant experiment into the following probabilities:
    1. What's the chance that your plant will survive the week?
    2. If it's dead when you return, what's the chance that your friend forgot to water it?
    3. If your friend forgot to water it, what's the chance that it'll be dead when you return?
    Let's now use a Table to solve for them:
    

    With that table we've answered the first of the three questions:

    1. The probabilitiy that your plant will survive the week (is not dead) is
    To get the second probability, we must use Bayes's Theorem (recall that the answer to the third question was actually given to us):
    
    1. The probabilitiy that your plant will survive the week (is not dead) is
    2. If it's dead when you return, then the chance that your friend forgot to water it is
    3. If your friend forgot to water it, what's the chance that it'll be dead when you return?

  • Now our author suggests that, rather than go through all this convoluted stuff, we think in terms of "natural frequencies". Let's see if we get the same answer any easier. Let's assume that there are 100 plants in the experiment, all either dead or not-dead, watered or not-watered.

    Once again, we'll use a Table. The trick to natural frequencies is to consider a large number of individual plants, and turn probabilities into numbers. Since the probability of not watering is .30 (30%), we might think "30 out of 100" wouldn't be watered. And that might suggest that we consider 100 plants:

    

    Now we fill in the rest of the table, using the info provided:

    1. 
    2. 
    3. 

• Next we consider the breast cancer problem:

  "Gigerenzer and his colleagues asked doctors in Germany and the United States to estimate the probability that a woman with a positive mammogram actually has breast cancer, even though she's in a low-risk group: 40 to 50 years old, with no symptoms or family history of breast cancer.

  "The probability that one of these women has breast cancer is 0.8 percent. If a woman has breast cancer, the probability is 90 percent that she will have a positive mammogram. If a woman does not have breast cancer, the probability is 7 percent that she will still have a positive mammogram. Imagine a woman who has a positive mammogram.

  "What is the probability that she actually has breast cancer?"

  We'll use "natural frequencies" to solve this problem. We begin by establishing the variables:

  • - has Breast cancer
  • - tests Positive

  then, of course, we have also defined the complements:

  • - doesn't have Breast Cancer
  • - tests Not Positive
  1. Let's turn the question into a conditional probablility:
     "What is the probability that she actually has breast cancer, given that she tested positive?"

     translates into

     

  2. To answer that question, let's start by considering the universe. The best way to represent it is as a "natural frequency" table.

     Here's what we know (turning the givens into probabilities):

     1. 
     2. 
     3. 
     

     Starting with 1000 women, let's fill in the rest of the table.

     Notice that our author estimates -- rounds -- answers, to make his life a little easier. You might check that our answers round off to his answers. Often the approximations are sufficiently good.

• Some reflections, based on the reading:
  • There are two entries of four in the table that we'd like to eliminate entirely: so-called "false positives" and "false negatives". Each has its problems. Characterize each type of problem in the breast cancer example.

    Characterize each type of error in terms of drug testing.

  • Now that we've calculated the exact values, what do you think of this quote from our text:

    "As for the American doctors, ninety-five out of a hundred estimated the woman's probability of having breast cancer to be somewhere around 75 percent."

    What are the consequences for your health care?

  • Notice the difference between the German doctors and the American doctors:
    • 33% (8 of 24) German doctors estimated less than 10% (close! -- 9%)
    • 95% of American doctors estimated around 75% (far! -- 9%)
    What do you make of that? Who gets more alarmed?

Links:

• Strogatz on-line: Chances Are
• Pascal handout