AMS 311 Joe Mitchell
PROBABILITY THEORY: Practice Exam 1
Some Possibly Useful Formulas:
P(Fj|E) = P(E|Fj)P(Fj)
Pn
i=1 P(E|Fi)P(Fi),∪iFi=S, Fi∩Fj=∅, i 6=j
Geometric(p): p(i) = (1 −p)i−1p, i = 1,2,....
Binomial(n, p): p(i) = n
ipi(1 −p)n−i, i = 0,1,...,n
Poisson(λ): p(i) = e−λλi
i!, i = 0,1,2,...
NegBin(r, p): p(n) = n−1
r−1pr(1 −p)n−r, n =r, r + 1,...
1. (6 points) You toss three fair dice. (a). (3 points) What is the probability that the three
dice do not all three show the same number? (b). (3 points) Describe precisely (in words, a
complete sentence) the sample space Syou used in part (a) Give an example of one element
of S, stating exactly what your notation means.
2. (12 points) Suppose that 30% of all houses need a paint job. Also, 15% of all houses need
both a paint job and a new roof. Further, 7% of all houses that need a new roof also need
new windows.
(a). (6 points) Let pbe the probability that a randomly selected house needs a new paint
job, given that it needs a new roof. What is the minimum possible value of p? (i.e., I want
you to find a number, α, so that you can show that p≥α(always), and that sometimes it
could be that p=α)
(Start by defining the events that you use and stating clearly what you know...)
(b). (6 points) Assume now that you are told that 50% of all houses that do not need a
new paint job do need a new roof. What is the probability that a randomly selected house
needs a new roof?
3. (12 points) Of the patients of a hospital, 30% of the redheads have had skin cancer, and
20% of the non-redheads have had skin cancer. Also, 40% of the patients are redheads.
(a). What is the probability that a random patient there has had skin cancer?
(Show your work! Define any events that you use, and be clear and neat!)
(b). Patient Joe has cancer; what is the probability that he is a redhead?
4. (10 points) Two of the cards of an ordinary deck of 52 cards are lost. What is the
probability that a random card drawn from this (defective) deck is a spade?
(Show your work! Define any events that you use, and be clear and neat!)
5. (15 points) I tell you that, for two events Aand B,P(A) = 4/5, P(B|A) = 1/2, and that
Aand Bare NOT independent. (They are dependent.) For each statement, say “True”,
“False”, or “Can’t Tell” (if you give some reasoning, you have a possibility of partial credit
if you are wrong).
(i). Aand Bare mutually exclusive
(ii). Aand A∩Bare independent
(iii). P(A) = P(A|B)
