Week 3
Discussion: Probability Paradox
A family member can go to one of two local hospitals for
heart surgery.
Checking the history for the past year, you find that each
of the two hospitals has performed cardiac surgery on 1000 patients. In
hospital A 710 patients survived (71%). In hospital B 540 (54%) survived.
Based on the numbers presented, which hospital do you think is
superior in cardiac surgery?
Surely hospital A is better, right?
Now, let’s look at more data. The below chart summarizes
three categories of patients (those entering in fair, serious and critical
condition) and the survival rate from surgery (in percent) for the two local
hospitals.
Patient Entering Condition
Hospital A
Hospital B
Survivors from A (# and percent)
Survivors from B (# and percent)
Fair
700
100
600 or 86%
90 or 90%
Serious
200
200
100 or 50%
150 or 75%
Critical
100
700
10 or 10%
300 or 43%
Total
1000
1000
710 or 71%
540 or 54%
Looking at the data broken down in this way, we see that
Hospital B has a higher success rate in all three categories of patients but
when averaged all together, Hospital A has the higher overall survival rate.
Based on the numbers presented, which hospital do you think is superior in
cardiac surgery?
Probability Puzzles
Please select one below to work on and share your answer.
Probability Puzzle 1: The Birthday Problem
There are 23 people in this class. What is the probability
that at least 2 of the people in the class share the same birthday?
Probability Puzzle 2: The Game Show Paradox
Let’s say you are a contestant on a game show. The host of
the show presents you with a choice of three doors, which we will call doors 1,
2, and 3. You do not know what is behind each door, but you do know that behind
two of the doors are beat up 1987 Hyundai Excels, and behind one of the doors
is a brand new Cadillac Escalade. The cars were placed randomly behind the
doors before the show, and the host knows which car is where. The way the game
is played out is as follows. The host lets you choose a door. Assume you choose
door #1. Before he opens door #1 to let you see what you have chosen, he opens
one of the remaining doors, say door #3, to reveal a Hyundai Excel (he will
always open one of the remaining doors that has the booby prize), and asks you
whether or not you want to change your choice to door #2. What do you tell him?
Probability Puzzle 3: Flipping Coins
If you flip a coin 3 times, the probability of getting the
sequence HTH is identical to the probability of getting HTT (1/8). Let’s make
this situation a little more interesting. Suppose you are going to flip a coin
until you get the sequence HTH. Say this takes you x flips. Then, suppose you
are going to flip the coin until you get the sequence HTT. Say this takes you z
flips. On average, how will x compare to z? Will it be bigger, smaller, or
equal?
Probability Puzzle 4: Disease Testing and False Positives
Assume that the test for some disease is 99% accurate. If
somebody tests positive for that disease, is there a 99% chance that they have
the disease?
Probability Puzzle 5: A girl named Florida
Here’s a three part puzzler:
1. Your friend has two children. What is the probability
that both are girls?
2. Your friend has two children. You know for a fact that at
least one of them is a girl. What is the probability that the other one is a
girl?
3. Your friend has two children. One is a girl named
Florida. What is the probability that the other child is a girl?
The Value of Variance
More often than not, when we are presented with statistics
we are given only a measure of central tendency (such as a mean). However, lots
of useful information can be gleaned about a dataset if we examine the
variance, skew, and the kurtosis of the data as well. Choose a statistic that
recently came across your desk where you were just given a mean. If you can’t
think of one, come up with an example you might encounter in your life. How
would knowing the variance, the skew, and/or the kurtosis of the data give you
a better idea of the data? What could you do with that information?
Example: Say you are
an executive in an automobile manufacturer, and you are told that, for a particular
model of new car that you sell, buyers have on average 2.2 warrantee claims
over the first three years of owning the car. What would additional information
on the shape of your data tell you? If the variance was low, you’d know that
just about every car had 2 or 3 warrantee claims, while if it was high you’d
know that you have a lot of cars with no warrantee claims and a lot with more
than 2.2. The skew would provide similar information; with a high level of
right skew, you’d know that the average is being brought up by a few lemons;
with left skew you’d know that very few of the cars have no warrantee claims.
The kurtosis (thickness of the tails) would help you get an idea as to just how
prevalent the lemon problem is. If you have high kurtosis, it means you have a
whole bunch of lemons and a whole bunch of perfect cars. If you have low
kurtosis, it means that you have few lemons but few perfect cars.
