C22.0103 Homework Set 3 Spring 2012
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1. The file CDCLifeTable is available from the course web site. It’s in the M folder;
the direct link is
http://people.stern.nyu.edu/gsimon/statdata/B01.1305/M/index.htm
This has data for non-Hispanic whites, with gender corresponding to separate sets of
columns.
a.
Plot the female and male yearly hazards (the q’s) as a function of age.
These should be overlaid on a single graph.
HINT: In using Minitab, select Graph  Scatterplot, and then
use “with connect line.” Set up the information panel like this:
Use Multiple Graphs to put both graphs on the same axis. You
can edit the labels if you wish by double-clicking on graph
features.
b.
Compute the gender ratio as a function of age. Let’s say you choose to do
this as “females per 100 males.” Plot this as a function of age.
HINT: In Minitab, use Calc  Calculator.
c.
Consider the marriage of Joseph, age 25, to Felicia, also age 25. Assume
that their future survival mechanisms operate independently (even though
this is a questionable assumption for a married couple). Find the
probability that
both are alive at age 75,
Joseph will be alive at age 75 but Felicia will have died
Felicia will be alive at age 75 but Joseph will have died
both die before age 75
2. Consider the following life table:
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Age (years)
Expectation of
Future Life (years)
30
27
25
22
20
19
18
17
16
15
14
13
12
11
10
9
7
5
Birth to 20
20 to 25
25 to 30
30 to 35
35 to 40
40 to 41
41 to 42
42 to 43
43 to 44
44 to 45
45 to 46
46 to 47
47 to 48
48 to 49
49 to 50
50 to 55
55 to 60
60 and up
This table is presented in terms of expected future lifetime, rather than the more common
(and more useful) x and dx notation. The Age column is given as intervals, but it’s easy
to link to the more conventional notation. For instance, think of the row “42 to 43” as
simply being “42.” This row refers to someone who has reached the 42nd birthday but not
the 43rd birthday.
Make a list of things that are wrong with this table. You need not do elaborate arithmetic
to form this list.
This is a genuine life table, and it was actually used!
3. Suppose that a pizza shop offers ten different toppings for its pizzas, and you select
three of these at random.
a.
b.
c.
d.
What is the probability that mushrooms will be among your selections?
What is the probability that both mushrooms and onions will be among your
selections?
What is the probability that your selections will be mushrooms, onions, and black
olives?
How exactly would you go about selecting three toppings at random?
4. Suppose that you are dealt five cards, at random, from a standard deck.
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a.
b.
What is the probability that exactly four of these cards will be spades?
What is the probability that at least four of these cards will be spades?
5. In games of the KENO or LOTTO style, the bettor selects numbers from a fixed set.
Then the game operator selects another set of numbers, and the bettor wins according to
the number of matches.
a.
Suppose that the game uses the numbers 1 through 50, and suppose that the
operator selects eight of these. If the bettor selects five numbers, find the
probability that there are exactly five matches. HINT: You might find the
 50 
number   = 536,878,650 to be useful.
 8
b.
Suppose that the game uses the numbers 1 through 50, and suppose that the
operator selects ten of these. If the bettor selects five numbers, find the
probability that there are exactly five matches. Also note whether this probability
is larger or smaller than the probability in a. HINT: You will want the number
 50 
  = 10,272,278,170.
 10 
c.
Suppose that the game uses the numbers 1 through 50, and suppose that the
operator selects ten of these. If the bettor selects six numbers, find the
probability that there are exactly six matches. Note whether the probability here
is larger or smaller than the probability in b.
HINT: You can do these with a calculator, of course, but the calculations
call be done more easily with Minitab’s hypergeometric option. The
command is Calc  Probability Distributions  Hypergeometric.
6. The random variable Y has the probability distribution described in this table:
y
P[ Y = y ]
1
0.30
2
0.40
3
0.20
4
0.10
Find the mean and standard deviation of the random variable Y.
7. The random variable Q has the probability distribution described in this table:
q
P[ Q = q ]
100
0.20
200
0.50
300
0.20
400
0.10
Find the mean and standard deviation of Q. (After problem 6, this should be very easy.)
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8. The random variable X has this distribution:
x
P[ X = x ]
37
0.06
38
0.25
39
0.38
40
0.25
41
0.06
Find the mean E X.
9. Suppose that you are playing a game of chance in which the probability of winning
is 0.14. What is the probability that you’ll win exactly once in six turns?
10. The admissions office of a small, selective liberal-arts college will only offer
admission to applicants who have a certain mix of accomplishments, including a
combined SAT score of 1,400 or more. Based on past records, the head of admissions
feels that the probability is 0.42 that an admitted applicant will come to the college. If
600 applicants are admitted, what is the probability that 250 or more will come? Note
that “250 or more” means the set of values {250, 251, 252, …, 600}.
11. Consider the admissions office in the previous problem. Based on financial
considerations, the college would like a class size of 250 or more. Find the smallest n,
number of people to admit, for which the probability of getting 250 or more to come to
the college is at least 0.95.
HINT 1:
HINT 2:
HINT 3:
Based on your solution to the previous problem, you see that n = 600 is
not large enough.
You will have to proceed by trial and error.
You want P[ X  250 ] to be 0.95 or larger. The organization of Minitab
makes it easier to ask for P[ X  249 ] to be 0.05 or less.
12. If X is the binomial phenomenon based on n trials and with success probability p,
then E X = np and SD(X) = standard deviation of X = np 1  p  . In the situation of
problems 8 and 9, with n = 650 and p = 0.42, find the mean and standard deviation of the
number of students who will come to the college.
14. Assume that P(A) = 0.7 and P(B) = 0.2.
a.
If A and B are mutually exclusive, find P(A  B).
b.
If A and B are mutually exclusive, find P(A  B).
c.
If A and B are independent, find P(A  B).
d.
If A and B are independent, find P(A  B).
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15. Andrea and Betty are making a wager over a sequence of coin tosses. Andrea wins
the first time there are two consecutive heads. Betty wins the first time tails-heads are
observed consecutively.
For example, these sequences are wins for Betty:
TH
TTTH
TTTTTH
What is the probability that Andrea wins this game?
16. The Sweet Novelty Company offers the special expensive ($150 each) Chocolate
Groundhog product. During January, the Internet orders for this very expensive item
come in at a rate of 1.2 per day, and it is believed that this phenomenon can be described
as a Poisson random variable with  = 1.2. Find
a.
the probability that there will be three or more orders on any day.
b.
the probability that, in a five-day work week, there will be at least one day
on which there are three or more order.
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