AMS 311, Lecture 9

February 22, 2001

Today’s review problem:

Cards are drawn at random from an ordinary deck of 52, one-by-one and without replacement. What is the probability that no heart is drawn before the ace of spades?

Definition A random variable is called Bernoulli with parameter p if its probability function is and and 0 otherwise.

What is E(X)? What is var(X).

Theorem 5.1

Consider a sequence where n is a fixed integer. We seek to describe the random variable Then the probability function for Snis given by

if and 0 otherwise.

What is E(Sn)? What is var(Sn)?

First Examination next Thursday (March 1). You will have assigned seating, and there will be multiple forms of the examination. Please bring a photo ID. You may have one bound document to refer to (either bound notes or bound text). There will be about eight problems of this approximate level of difficulty:

Some suggested problems for you to check yourself with.

  1. Consider two events, A and B. Let P(A)=0.4, and let P(B)=0.8. Use a Venn diagram to help you visualize the following?
  2. What is the maximum possible value for P(AB)?
  3. What is the minimum possible value for P(AB)?
  4. What is the maximum possible value for P(ABc)?
  5. What is the minimum possible value for P(ABc)?
  6. You toss three fair dice independently. What is the probability that the three dice do not all show the same number?
  7. A five member committee is selected at random from a class consisting of 23 women and 19 men. What is the probability that there is at least one man on the committee?
  8. A student is about to take a 100 question multiple choice test. Each question has 5 choices. If the student knows the correct answer, he marks it correctly with probability 0.95 (there is a 5 percent chance that he slips up and mismarks the choice, even though he knows the correct answer); otherwise he guesses at random. Assume that the student has correct knowledge of 70 percent of the course material.
  1. On what fraction of the questions will he mark the correct choice?
  2. Suppose the student gets question 27 correct. What is the probability that the student actually knew the answer?

Questions from one form of a past examination.

  1. Let A, B, and C be three events. Show that exactly two of these events will occur with probability
  1. Let A be an event such that P(A)=0.5, and let B be an event such that P(B)=0.8. What is the maximum possible value of P(AB) and the minimum possible value of P(AB)?
  2. AThe Great Carsoni,@ a magician, claims to have extrasensory perception. In order to test this claim, he is asked to identify the 4 red cards out of 4 red and 4 black cards which are laid face down on the table. AThe Great Carsoni@ correctly identifies 3 of the 4 red cards. Thereafter, he claims to have proved his point. What is the probability that AThe Great Carsoni@ would have correctly identified exactly 3 of the 4 red cards if he were, in fact, guessing?
  3. A fair die is rolled 7 independent times. What is the probability that at least one face will not appear?
  4. A box contains 8 good and 2 defective items. Four items are selected at random without replacement from the box. What is the probability of finding at least one of the defective items?
  5. The probability that a randomly selected individual has a specified medical condition is .05. A screening test for this condition has sensitivity .99. That is, the probability that an affected individual tests positive is .99. The specificity of the procedure is .95. That is, the probability that an individual not affected tests negative is .95. What is the probability that the screening test will produce a false positive reading? That is, given that the results of the test are positive, what is the probability that the individual does not have the condition?
  6. In poker, a flush consists of 5 cards of the same suit. Given that all of the cards in a randomly selected poker hand (that is, a draw of 5 cards) are red (either hearts or diamonds), what is the conditional probability that the hand is a flush?
  7. Each day an experimental animal is exposed to a certain set of stimuli designed to elicit a particular response. Let Ak be the event that the animal makes the desired response on the kth day, and suppose that P(Ak+1|Ak)=β and that P(Ak+1|Akc)=α, where 0<α<β1. Let pk=P(Ak).

Show that pk+1=α+(β-α)pk.