Cross Product Sample Spaces 49115. Suppose A and B are events in a sample space such that P(A) = 1/4, P(B) = 5/8,
and P(A U B) = 3/4. What is P(A n B)?
- In a certain group of people, 50% are right-handed and wear glasses, 5% are left-
handed and wear glasses, and 1% are ambidextrous and wear glasses. What is the
probability that a person selected at random from this group wears glasses? Assume
that ambidextrous means neither right-handed nor left-handed but, rather, some mix-
ture of both. In particular, the ambidextrous people are not included in the set of right-
handed people or in the set of left-handed people. - Four cards are dealt at random from a deck. What is the probability that at least one
of them is an Ace? The answer may be given in terms of the combinatorial notation
C(a, b). - In a fierce battle, not less than 70% of the soldiers lost one eye, not less than 75% lost
one ear, not less than 80% lost one hand, and not less than 85% lost one leg. What is the
smallest percentage who could have lost simultaneously one ear, one eye, one hand,
and one leg? This problem comes from Tangled Tales by Lewis Carroll, the author of
Alice in Wonderland. - The waiting room of a dentist's office contains a stack of 10 old magazines. During
the course of a morning, four patients, who are waiting during non-overlapping times,
select a magazine at random to read. Calculate in two ways the probability that two or
more patients select the same magazine. - What is the probability that in a group of 10 people, at least 2 have the same birth-
day? Assume that nobody was born on February 29th. Use a calculator to get a good,
approximate answer. - Suppose El, E 2 ... , En are events (not necessarily disjoint) in a sample space Q2
endowed with a probability density p. Find an expression for P(U 1 <i<n Ei), and prove
that your expression is valid. (Hint: Make an analogy to the Principle of Inclusion-
Exclusion of Section 1.5, but add up probabilities instead of elements.)- Recall that by definition, a discrete sample space may contain a countably infinite
number of outcomes. This exercise gives an example of such a countably infinite
sample space. Suppose we flip a fair coin until it comes up heads. Of course, there is
no way to know in advance how many flips will be required. Design a sample space
and a probability density to model this situation. Prove that the probability density you
define is legitimate.
W Cross Product Sample Spaces
Many probabilistic situations involve repeating an experiment over and over or combin-
ing the results of several unrelated experiments. Repeated coin flipping and the Birthday
Problem are two examples of such situations. In both cases, it is appropriate to choose a
sample space of n-tuples; the ith component, where i ranges from 1 to n, represents theoutcome of the ith flip-or the birthday of the ith person. These are both examples of
cross product sample spaces, the subject of this section. If the coin is fair, and if the days
of the year are equally likely, then we assign a uniform probability density. In many situa-
tions, however, it does not make sense to do this (the coin might be biased, for example).
This section explains how to assign reasonable probabilities in such situations. We study in