530 CHAPTER 8 Discrete Probability
- Repeat Exercise 3 for a coin that comes up heads a third of the time.
5. What is the relationship between b(k; n, p) and b(n - k; n, p) when p = 1/2? Does
this relationship hold if p # 1/2? - Suppose we make draws from an urn containing two red balls and three black ones,
replacing the chosen ball after each draw. How many draws should we make (we have
to decide this number in advance) to have probability 0.5 or greater of selecting at least
two red balls?
7. Suppose we draw three balls from an urn containing two red balls and three black
balls. We do not replace the balls after we draw them. In terms of the hypergeometric
distribution, what is the probability of getting two red balls? Compute this probability.
8. Compute the expectation E(X) of the random variable X that counts the number of
heads in four flips of a fair coin. (See Exercise 3.)
9. Compute the expectation E(X) of the random variable X that counts the number of
heads in four flips of a coin that lands heads with frequency 1/3. - Suppose we make three draws from an urn containing two red balls and three black
ones. Determine the expected value of the number of red balls drawn in the following
situations.
(a) The chosen ball is replaced after each draw.
(b) The chosen ball is not replaced after each draw. - We have seen two ways to compute the expected value E (X) of a random variable X.
One way is to use the definition of expected value, summing px (x) over the range QŽx
of X. The other way is to use Theorem 2, summing X (w) • p(w) over the domain Q2
of X. If you have not already done so, do Exercise 10, and then compute E (X) using
the method you did not use the first time. In general, which method do you think will
be easier to carry out, and why? - Game A has you roll a fair die once and receive the number of dollars that is equal to
the value on the top face. Game B has you roll a fair die twice and receive the number
of dollars that is the maximum of the two values that show on the top face. It costs $3
to play game A and $4 to play game B. Which game would you choose? - Wagga Wagga University has 15,000 students. Let X be the number of courses for
which a randomly chosen student is registered. No student is registered for more than
seven courses, and each student is registered for at least one course. The number of
students registered for i courses where 1 < i < 7 is 150, 450, 1950, 3750, 5850, 2550,
and 300, respectively. Compute the expected value of the random variable X.
W Variance, Standard Deviation, and the Law of Averages
The actual value X (a)) of a random variable X can differ drastically from its expected
value E(X). To measure how well the values of X tend to cluster around its expectation
E(X), we define in this section the notions of variance and standard deviation of a random
variable.
Suppose we repeatedly run an experiment, and suppose each time we compute the
average of a certain set of random variables. We do not anticipate that we will get the same
number for the average each time, but we do wonder how this number will behave. Will
we see wild fluctuations in the number from one run of the experiment to the next? This
section studies this issue, and it concludes with a technical version of a theorem known