1.  If A is an event, let A^c represent the complement of that event.  You know that P(A^c) = 1- P(A).  So if you know that the probability of a medical test
showing a positive result if you really have the disease is 95%, is it true that the probability of the test showing a positive result when you don't have the disease is 5%?  Explain.

2. A recent study by the American Society of Witches has determined that approximately 2% of the U.S. population are witches.  Witches are hard to identify, because they no longer wear the pointy hats, and plastic surgery has eliminated the long, pointy noses.  However, 90% of all witches have warts on their noses.  In all fairness, though, people who are not witches can also have warts.  In fact, 15% of all non-witches have warts on their noses.  Suppose that you spot a person with a wart on their nose.  What's the probability the person is a witch?

3. DNA "fingerprinting" claims to be able to use traces of the DNA found in blood left on the crime scene to accurately identify criminals.  Basically, DNA fragments from the crime scene are compared with fresh samples from the suspect himself.  Of course, the crime-scene fragments have sometimes degraded over time, making identification more complicated.  A match is declared if a "fingerprint" from the crime scene (which looks a little like a bar-code) looks like the fingerprint from the person.  Because of slight variations in the lab and because of the degredation of the sample from the crime scene, these "bar codes" rarely match exactly, even if from the same person, and so some subjective judgement is made.  If the two samples (crime sample and "fresh" sample) are really from the same person, a match will correctly be declared about 99.9% of the time.  If the samples are in fact from different people, a match might still be mistakenly declared, and this happens about .3% (0.003) of the time.  (These numbers are hypothetical.)

Suppose a murder is committed in a small city of 500,000 people.  If we arrested someone randomly , the probability that this person committed the crime is 1/500000.  Suppose, now, that you are sitting on a jury.  The prosecutor tells you that the defendant is guilty of this crime "beyond all reasonable doubt."  The evidence of his guilt is that his DNA finger print matched the DNA fingerprint at the crime scene, and this test is 99.9% accurate.  "There is only a .1% chance that we've made a mistake!" he says.  The Defense attorney, on the other hand, says that this is a misleading comment, because the probability that they've falsely declared a match is .3%, not 100 - 99.9%.  "Furthermore," the defense says, "While a .3% chance of an error may seem small to you, in a city of 500,000 people this means that there are 1500 people who could be falsely matched and facing a jury trial to save their lives and their reputations.  So .3% is not "beyond a reasonable doubt.""

(HINT: If there is a true match, then the defendent is guilty.  )

a) Is the prosecutor correct to say that the probability they've caught the wrong person is .1%?  Explain.
b) What is the probability that, if a match is declared, the two samples really are from the same person?

SKIP NUMBER 4!
4. Let X represent the change of weight a person will lose in one week when put on a special diet.  A researcher models this random variable as having a uniform distribution on the interval [-5, 2].  Assuming this model is correct:
a) Is X a continuous or discrete random variable?
b) What's the probability a person will lose weight?
c) What's the probability that a person will lose between 0 and 3 pounds?
d) What's the probability a person will lose more than 4 pounds?
e) If infinitely many people took this diet, 10% of them would lose more than x pounds.  What's x?

5. A jar contains 10 candies.  3 are Red, 2 are Brown, 5 are Green.  Remove two candies without replacement.  Let X = number of red candies removed. Write the probability density function (as a table) for X.

6. Toss a coin 10 times.  What's the probability of getting 1 or more heads?

7.  A coin is tossed 10 times.  Which outcome is the least likely?  Explain:
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8. A roullette wheel is a horizontal wheel with slots along the outer rim labelled  00,0,1-36.  The wheel is spun and a ball is dropped into the center.  The ball is equally likely to land on any of these numbers.  One bet you can place is "Even", which is a number between 2-36.  Bets on "odd" or "even" pay even money, which means that if you bet a dollar, you can either lose a dollar, or win a dollar.  Statisticians are optimists, and rather than say "Lose 1 dollar" we say "Win -1 dollar."  Suppose you place a one dollar bet on "Even."
a) What's the probability you will win 1$?
b) Suppose that 20 spins in a row come up "Even".  Your friend tells you that you should bet "Even" again because "Even" is "hot."  Explain why this is incorrect.  What is the probability that the next spin will be "Even"?
c) Suppose the wheel is spun twice, and each time you place a 1$ bet on "Even."  Let Y represent your total winnings. Write the probability density function for Y.