MATH 183 - PRACTICE EXAM #1 - Corrected
closed book, no calculators, no headphones
Each problem is worth the same number of points
1) Define and give an example:
a) the 4 axioms for probability P(A), A = subset of sample space; p. 36
b) Conditional Probability P(A|B) for subsets A,B of the finite sample space S; p. 43
c) A,B independent events in the sample space S; p. 70
d)
nPk,
nCk=
⎛n⎞
⎜ ⎟ ; pages 93, 107
⎝k ⎠
e) the binomial theorem; p. 108
f) random variable, discrete or continuous; p. 129, paragraph 5
g) probability density function of a random variable X; discrete on p. 155; continuous on p. 168. Give
properties.
h) mean or expected value of a random variable X; p. 175; E(g(X)), p. 186
i) variance of a random variable X; p. 194
j) binomial probability distribution (give the pdf); p. 131
k) hypergeometric probability distribution (give the pdf); p. 139
l) cumulative distribution function; discrete, p. 159; continuous, p. 170
2) LNG (liquefied natural gas) is to be transported by tanker ship. Using data from tankers carrying less
dangerous cargo, it is estimated that the probability is 8/50,000 that an LNG tanker will have an accident on
any one trip. It is also estimated that given an accident has occurred, the probability is 3/15,000 that the
damage will be bad enough that an explosion will happen. What is the probability that a given LNG tanker will
cause a catastrophic disaster? Note: this is a conditional probability problem. p. 46.
3) Suppose that you have a string of 24 Christmas tree lights wired in series. The chance of a bulb working
the first time current goes through is thought to be 99.9%. What is the probability that the string does
not work? Assume each bulb independent of the rest. p. 78
4) Find the probability that all 70 students in a class have different birthdays. What is the probability that
2 or more students in that class have the same birthday? p. 117
5) Doomsday airlines has 2 planes (an old 2 engine prop plane and another old 4 engine prop plane). Each plane
will land safely only if at least half its engines are working properly. Assume that each engine on the plane has
the same probability p of failing and that such failures are independent events. Find
P(flight lands safely for the 2-engine plane)
P(flight lands safely for the 4-engine plane).
For what values of p should you use the 2-engine plane because it is more likely to land safely? p. 135
6) An urn has 9 chips, 6 are black and 3 are purple. Suppose you draw 6 chips without replacement. What is
the probability that those 6 will include at least twice as many black as purple chips? p. 147
7) Suppose that X is a continuous random variable and that a,b are constants with a≠0. Show that the pdfs
of X and Y=aX+b are related by the formula
⎛ u − b ⎞ |a|-1.
fY (u ) = f X ⎜
⎟
⎝ a ⎠
8) Find the expected value of the exponential distribution with pdf
p. 171
fY (u ) = λ e − λu , for y>0 and 0 otherwise.
Then find the variance of this distribution. p. 185, 199.
9) Show that Var(X) = E(X2)-(E(X))2. p. 195 for continuous case. Do the discrete case too.
10) A fair die is rolled 3 times. Let X be the number of different faces showing (X = 1,2 or 3). Find E(X). p.
186.