Stat 330
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Homework 2
Spring 2009
Kolmogorov
Let A, B ⊂ Ω. Draw a Venn diagram that shows the sets A ∪ B, A ∩ B, and A ∩ B̄. (Many drawings are
correct, depending on your choices of A and B.) Then, use Kolmogorov’s axioms to show that the following
statements are true.
(a) For any two evens A, B ⊂ Ω,
P (A ∩ B̄) = P (A) − P (A ∩ B)
(b) For two events A, B ⊂ Ω with B ⊂ A,
P (A ∩ B̄) = P (A) − P (B)
(c) For any two events A, B ⊂ Ω,
P (A ∪ B) = P (A) + P (B) − P (A ∩ B)
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Broken Components
Four components are inspected and three events are defined as follows:
A : “all four components are found to be defective”
B : “exactly two components are found defective”
C : “at most three components are found defective”
Define a suitable sample space Ω. Then, interpret the following events. (In other words, express the events
below in plain English).
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a) B ∪ C.
b) B ∩ C.
c) A ∪ C.
d) A ∩ C.
Events and Notation
Suppose that A, B, and C are three events in an expriment. Express each of the following events in set
notation.
(a) At least one of the three events occurs.
(b) None of the three events occurs.
(c) Exactly one of the three events occurs.
(d) Exactly two of the three events occurs.
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Counting & Probability
For all of the following problems, find the sample space first. Determine the size of the sample space, then
deal with the specified event.
(a) A lottery has 53 numbers from which seven are selected without replacement. You play the lottery by
selecting seven numbers from the same 53 numbers without replacement. What is the probability that
your seven numbers are the same as the lottery’s?
(b) Find the probability of being dealt three kings in a five-card hand in a 52-card standard deck (no
jokers) when the cards are drawn without replacement? What if the cards are drawn with replacement
(and re-shuffling between each draw)?
(c) How many different passwords of length 6 can be generated from the set of letters ’a’-’z’, ’A’-’Z’, and
the digits ’0’, ’1’-’9’ using at least two digits? Symbols can be used more than once.
(d) How many ways are there to arrange the letters M I S S I S S I P P I to different ”words” of length
11? Why is it not just 11! ?
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Deriving Probabilities
(a) Restaurant
A certain restaurant has two cooks—the chief cook and his assistant. On any given day, the probability
that the chief cook will show up for work is 0.97, the probability that the assistant cook will show up
for work is 0.96, and the probability that at least one of the two cooks will show up for work is 0.98.
Find the probabilities that on any given day the restaurant will have
a) both cooks,
c) only the chief cook,
e) only one of the two cooks.
b) neither of the two cooks,
d) only the assistant cook,
(b) Best-By
There are five containers of milk on a shelf; unknown to you, two of them have passed their use-by
date. You grab two at random. What’s the probability that neither have passed their use-by date?
Suppose someone else has got in just ahead of you, taking one container, after examining the dates.
What’s the probability that the two you take after that are ahead of their use-by dates?
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