Day 06, MAT115

• Announcements:
  • You have a new probability assignment, due Friday. Come with questions about it on Wednesday (in other words, start early!).

• Last time we talked Monty Hall, and fish, and Pascal. All of them require counting.

  Your reading for today (hopefully you've read it) is another attempt to convince you that probability is all about counting.

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

  Another thing 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.

• We'll start with an application of Pascal's triangle, which we looked at last time. Here's how we can think of forming Pascal's triangle:
  

  and here's the result of doing that over and over:

  

  How do we use it to carry out probability problems?

  Last time we saw how the triangle can be used to count the universe of possible events. We looked at the row

  1 5 10 10 5 1

  and saw how we could use it to answer the question

  "How many ways are there of putting 5 people into 2 cars?"

  We can interpret the numbers in that row as the number of ways of putting

  • all in car one (and none in car two);
  • 4 in car one, and 1 in car two;
  • 3 in car one, and 2 in car two;
  • 2 in car one, and 3 in car two;
  • 1 in car one, and 4 in car two;
  • none in car one (and all in car two)
  We see the beautiful symmetry in the row from Pascal's triangle, and the beautiful symmetry in the description of what it's counting in terms of cars.

  So here are some typical probabilities, based on that row of Pascal's triangle:

  1. In how many distinctly different ways can we put 5 people into 2 cars? (this is the size of the universe)
  2. What's the probability that we put 2 in car one, and 3 in car two? (now we're counting: that is, counting using Pascal's triangle)
  3. What's the probability that we put everyone one in a single car? (count again -- and it's figuring out which occurrences to count!)
  4. What's the probability that no one is alone in a car? (use the complement!)

• Now on to "Chances Are":

  This reading concerns "conditional probability", which is the probability of an event given that some other event has already occurred. So it's sort of a "probability within a probability" problem, and that just seems to make them twice as hard! But bear with me, and we'll see if we can't follow Strogatz as he attempts to turn what seems difficult into something reasonable, through "right thinking".

  Well, right away our author does something regrettable -- he says that "...the probability that your friend will forget to water it is 30 percent." That's the chance that your friend will forget to water it, right? The probability is .30. So I want you to convert throughout: when Strogatz gives a probability as a percentage, you convert it to a decimal (or fraction).

  Let's consider "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

  • Let's start by considering the universe. We'll represent it in a slightly different way, however, using a table. The first problem to consider is the problem of the plants, living or dead, and the watering (done or not). Because there are these two ways of looking at the situation of our plants and their waterings, our universe will be divided into two separate worlds....
    1. Venn diagrams, and
    2. a Table.
    Let's look at the Venn diagram for our universe first, because that will introduce us to the idea (and notation) of intersection: when two events happen simultaneously.

  • Next we need to introduce this notion of conditional probabilities. They're a kind of "side bet". You reduce your focus to just a part of the universe, the part that has already come to pass. Here's how we write them: for example,
    The robability of given

    is written as

    

    Now our author mentions "Bayes's Theorem", and, just so you know what that is, here it is. It 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. 
    Can you read them? Can you turn those probabilities into English?

  • 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:
    

  • 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.

• 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.

  1. Let's start by considering the universe. The best way to represent it is as a table.
  2. Then we write down our givens:
     1. 
     2. 
     3. 

  3. Then we solve for the unknown, which is

     We can also fill in the rest of our table, and do other probabilities.

• Notice that our author estimates -- rounds -- answers, to make his life a little easier. We'll try that first, then calculate the exact probabilities. Often the approximations are sufficiently good.

• Some reflections, based on the reading:
  • 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