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 p(0)  1  p and p(1)  p and 0 otherwise.
What is E(X)? What is var(X).
Theorem 5.1
Consider a sequence X 1 , X 2 , , X n where n is a fixed integer. We seek to describe the
random variable S n 
n

i 1
X i . Then the probability function for Sn is given by
p( x)  Pr( Sn  x) 
if x  0,1, 2,, n and 0 otherwise.
 p
n
x
x
(1  p) n x ,
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?
a. What is the maximum possible value for P(AB)?
b. What is the minimum possible value for P(AB)?
c. What is the maximum possible value for P(ABc)?
d. What is the minimum possible value for P(ABc)?
2. You toss three fair dice independently. What is the probability that the three
dice do not all show the same number?
3. 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?
4. 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.
a. On what fraction of the questions will he mark the correct choice?
b. 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
P(AB) + P(AC)+ P(BC) - 3P(ABC).
occur with probability
2. 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)?
3. 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?
4. A fair die is rolled 7 independent times. What is the probability that at least
one face will not appear?
5. 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?
6. 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?
7. 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?
8. 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.