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Unformatted text preview: 22M10 Exam 2, Fall 1995', Camillo. N o calculators.
Show all work for full or partial credit. 1.(10) You roll two dice, what is the probability that the total is at least 11?
2.(10) You deal four cards. How many such hands do not contain a red card? 3.(10) You chose two numbers from the numbers 1 through 10, including 1 and 10. What
is the porobability that both numbers you chose are odd?. 4.(10) How many committees of three people can be chosen from a club with six members? 5.(25) A survey of 100 students counted:
a. The number who had purple hair. b. The number who had blue shoes. c. The number who had orange sunglasses. It was found that : None had all three. One had purple hair and orange sunglasses but not blue shoes. One had blue shoes and orange sunglasses but not purple hair. One had purple hair but did not have blue shoes and did not have orange sunglasses.
Three had blue shoes. Three had purple hair. Three had orange sunglasses. How many had neither purple hair, nor blue shoes nor orange sunglasses? 6.0.5) A hospital nutritionist prepares a menu for patients on low fat diets. The primary
ingredients are chicken and corn. The meal must contain at least forty  ﬁve grams of
protein and at least ﬁfteen milligrams of iron. Each three ounce serving of chicken contains
forty —ﬁve grams of protein, nine milligrams of iron and four grams of fat. Each one cup
serving of corn contains nine grams of protein, six milligrams of iron and two grams of fat.
How much chicken and corn should be included'in the meal in order to minimize the
amount of fat? I 7.(10) 'If E and F are events such that Pr(E)=.5, Pr(F)=.5 and Pr(EﬂF)=.2, find
Pr{(EUF) }. ‘  ZZMIO Exam 2, Fall 1996, Camilla. No calmilators.
Show all work for full or partial credit.  l. (15) A box contains three slips of paper, one red, one yellow and one blue. You
withdraw slips from the box ten times with replacement. What is the probability that you draw red three times, yellow three times and en four times?
(,LQ._ 2. (15) E and F are events with the following properties. The probability E will happen is .5, the probability that F will happen is .3 and the probability that neither will happen is .3. What is the probability that both will happen? That is, compute )2: J
0 3. (10) You chose two numbers from the numbers 1 through 10, including 1 and 10 withoat
replacement. What is the probability that both numbers you chose are less than four? 56 4. (IS) A shipment of refrigerators contains two defectives. [f you select three what is the
probability that you chose at least one defective? 5. (15) A family consists of two parents, three girls and three boys. The family lines up for
a photograph. What is the probability that the three girls are standing together? 6. (15) Assume all days of the week are equally likely for birth occurances. [f four people
are in a room, what is the probability that no two were born on the same day? 7. (15} You walk from A to B in the picture below by going only east and south. What is
the probability you do not walk through C? “mmwmmp—wmmmmmmmem L. ’ Z 3 L/ 67 é 651””qu 1/96
1 L 5 Cl; 5 Q 7 ‘
2. 3 ‘2’ 5 é 7 3/
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’7‘ '5 5% /2, .
5 [email protected];’2 7/2 7;, C/J 3/)é’, 7/ 3,01] /O§ (6,42) W 40 CW 61? wotwh'ialr«Iﬂ:1f1‘H"!t"1ri"~’>"I" FIGURE 1 O 17. Let E and F be events such that  Pr(E) : .3, ?r(F') : .6. and _Pr(E or, :01,
What is Pr(E rt F)? ' ' ' . in a certain manufacturing process'the probability of a type l defect is .l2, the
a probability of a type ll defect is .22, and the probability of having both types of
defects is .02. Find the probability of having neither type of defect. [9. A man has six different pairs of socks. from which he selects two socks at random.
711' What is the probability that the selected socks will match? 20. Of the [5 members on a Senate committee. to plan to vote “yes" and 5 plan to
0 vote no' ‘on an important issue. A reportei attempts to ptcdict the outcome of the 7/ vote by questioning 6 of the senators. Find the probability that this sample ts precisely representative of the ﬁnal vote. That is ﬁnd the probability that four of the six
senators questioned plan to'vote yes. '  21. A man. a woman, and their three children randomly stand in a row for a family
52/; picture. What is the probability that the parents will be standing next to each other? 22. [n poker the probabilities of being dealt a ﬂush, a straight. and a straight ﬂush are *CD 5? 1‘l'cs'OOI‘JEiStl, .0039246. and .0000l54,'respectively. What is the probability of being dealt a straight or a ﬂush? 23. Figure 1 shows a partial map of the streets in a certain city. A tourist starts at point
A and selects at random a path to point B. (We shall assume that he walks only [5" south and east.) Find the probability that. #(a) He passes through point C. 1232: (b) He passes through point D.
:1 (c) He passes through point C and point D. r 97401) He passes through point C or point D. ' 2J4 A couple decide to have four children. What is the piobability that they will have
% more girls than boys?  25 A law ﬁrm' has six senior and fourjttnior partners. A committee of three partners
is selected at randorn’ to represent the ﬁrm at a conference. What is the probability
1.9 that at least one of the junior partners is on the committee? I; A coin is to be tossed six times. What is the probability of obtaining exaCtly three
' heads? ' 27 A bag contains nine tomatoes of which one is rotten. A sample of three tomatoes
[6 is selected at random. What is the probability that the sample contains the rotten tomato? 28. A vacationer has brought along fottr novels and fOur nonﬁction books. One day the
‘3! person selects two at random to take to the beach. What is the ptobabiiity that both I are novels? . t
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i 22M:10 Exam 2 Practice Problems I. A box contains three slips of paper, one red, one
yellow and one blue. You withdraw slips from the
box ten times with replacement. What is the
probability that you draw red three times, yellow
three times, and blue four times? 2. E and F are events with the following properties.
The probability E will happen is .5, the probability
that F will happen is .3, and the probability that
neither will happen is .3. What is the probability
that both will happen? That is, compute P(EnF). 3. You chose two numbers from the numbers 1
through 10, including 1 and 10 without replacement.
What is the probability that both numbers you chose
are less than four? 4. A shipment of 50 refrigerators contains two
defectives. If you select three what is the
probability that you chose at least one defective? .5. A family consists of two parents, three girls and
three boys. The family lines up for a photograph.
What is the probability that the three girls are
standing together? 6. Assume all days of the week are equally likely
for birth occurrences. If four people are in a room,
what is the probability that no two were born on the same day? 7. You walk from A to B in the picture be10w by
going only east and south. What is the probability
you do not walk through C? E") 8. Youroll two dice. What is the probability that
the total is at least 11? 9. You deal four cards. How many such hands do not
contain a red card? 10. You chose two numbers from the numbers 1
through 10, including 1 and 10. What is the
probability that both numbers you chose are odd? 1 1. How many committees of three people can be
chosen from a club with six members? 12. Of the 15 members on a Senate committee, 10
plan to vote “yes" and 5 plan to vote "no" on an
important issue. A reporter attempts to predict the
outcome of the vote by questioning 6 of the senators.
Find the probability that this sample is precisely
representative of the final vote. That is, find the
probability that four of the six senators questioned plan to vote “yes." 13. A hospital nutritionist prepares a menu for
patients on low fat diets. The primary ingredients
are chicken and corn. The meal must contain at least
fortyﬁve grams of protein and at least ﬁfteeri
milligrams of iron. Each three ounce serving of
chicken contains fortyfive grams of protein, nine
milligrams of iron, and four grams of fat. Each one
cup serving of corn contains nine grams of protein,
six milligrams of iron and two grams of fat. How
much chicken and corn should be included in the
meal in order to minimize the amount of fat? 14. If E and F are events such that P(E) = .5,
P(F) : .5, and P(EnF) = .2, ﬁnd P(EUF). 15. Let E and F be events such that P(E) = .3,
P(F') = .6, and P(EuF) = .7. What is P(EnF)? 16. In a certain manufacturing process the
probability of a type 1 defect is .12, the probability
of a type II defect is .22, and the probability of
having both types of defects is .02. Find the
probability of having neither type of defect. 17. A survey of 100 students counted: C
a) the number who had purple hair 13) the number who had blue shoes c) the number who had orange sunglasses [t was found that: None had all three. One had purple hair and orange sunglasses but not
blue shoes. ,. _ One had blue shoes and orange sunglasses but not purple hair. .
One had purple hair but did not have blue shoes and did not have orange sunglasses.
Three had blue shoes. Three had purple hair. Three had orange sunglasses. How many had neither purple hair, nor blue shoes
nor orange sunglasses? 18. A man has six different pairs of socks, from
which he selects two socks at random. What is the
probability that the selected socks will match? 19. A man, a Woman, and their three children
randomly stand in a row for a family picture. What is the probability that the parents will be standing
next to each other? 20. In poker the probabilities of being dealt a flush,
a straight, and a straight ﬂush are .0019654,
.0039246, and .0000154, respectively. What is the
probability of being dealt a straight or a ﬂush? 21. In the ﬁgure, a partial map of the streets in a
certain city is shown. A tourist starts at point A
and selects at random a path to point B. (We shall
assume that he walks only south and east.) Find the
probability that: a) He passes through point C. b) He passes through point D. c) He passes through point C and point D.
d) He passes through point C or point D. 22. A couple decide to have four children. What is the probability that they will have more girls than
boys? 23. A law firm has six senior and four junior
partners. A committee of three partners is selected
at random to represent the ﬁrm at a conference.
What is the probability that at least one of the
junior partners is on the committee? 24. A coin is to be tossed six times. What is the
probability of obtaining exactly three heads? 25. A bag contains nine tomatoes, of which one is
rotten. A sample of three tomatoes is selected at
random. What is the probability that the sample
contains the rotten tomato? 26. A vacationer has brought along four novels and
four nonﬁction books. One day the person selects
two at random to take to the beach. What is the
probability that both are novels? ...
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