II Exam - Answers to Review Sheet
MATH 105, Spring 2005
1. A committee contains 8 women and 4 men. They wish to choose a subcommittee of 4 members. If this subcommittee is chosen at random, what is the probability that: (a) All Men:
C(4,4) C(12,4)
=
1 495
.002 =
168 495
(b) Two of Each:
C(8,2)C(4,2) C(12,4)
.339 1 .002 = .998
(c) At least one woman: 1 (d) At least one of each: 1 (e) Pam on the committee:
C(4,4) C(12,4) C(4,4) C(12,4)
C(8,4) C(12,4) 165 495
=1
1 495
70 495
=
424 495
.857
C(11,3) C(12,4)
=
.333
2. An urn contains 12 numbered balls: 8 red and 4 blue. Three of the red balls have even numbers, and 1 of the blue has an even number. The rest of the balls have odd numbers. (a) P r[red] =
8 12
=
2 3
3 12 4 12
(b) P r[red|even] = (c) P r[odd] =
8 12
=
3 4
=
2 3
3 12 4 12
(d) P r[odd|blue] =
=
3 4
(e) No - both probabilities change 3. In the experiment above, suppose that you pay $1.00 to draw a ball. An odd numbered red ball gets you nothing, an even numbered red pays $1.00 back, an odd numbered blue gets $5.00, and the even numbered blue gets you $10.00. What is the expected value? Value -1 0 4 9 Probability 5/12 3/12 3/12 1/12 Total: Product 5/12 0 12/12 9/12 16/12 1.33
4. A widget factory creates two types of widgets: type A and type B. Type A widgets are defective 5% of the time, and type B widgets are defective 10% of the time. If the factory produces 20 type A and 60 type B widgets in a given day, and a randomly selected widget is defective, what is the probability that it is type A? If a randomly selected widget is defective, not what is the probability it is type B?
.05
D
.0125
A
20/80 .95
N
.2375
(a) P r[A|D] = (b) P r[B|N] =
.0125 .0125+.075
.4129 .7400
60/80
.10
D
.075
.676 .2375+.676
B
.90
N
.676
5. A student takes a multiple choice test made up of 10 questions. Each question has ve possible answers, one of which is correct. (a) P r[6 correct] = C(10, 6) (b) (c)
1 6 4 4 .006 5 5 8 4 2 P r[at least 8] = C(10, 8) 1 + C(10, 9) 5 5 1 Expected Score = np = 10 3 = 3.3 1 9 5 4 1 5 1 10 5 4 0 5
+ C(10, 10)
.000078
(d) Normal Distribution with = np = 50
x
score of 15 gives a Z-score Z = = from the table, the area above 1.77 is .5 .4616 = .0384. So, the probability of scoring 15 or more is only .0384, or 3.84%. 6. Answers follow
1 5 = 10 and 3015 2.83 1.77 and
=
npq =
30
1 5
4 5
2.83, the
5 4 3 2 1
(a)
6
9
12
15
18
(b) Median: 12, Mode: Trimodal with 11, 12, and 18. (c) Mean: X =
15+11+18+6 4
= 12.5, StDev: S =
1...