UNIVERSITY OF MINNESOTA, MORRIS

INTRODUCTION TO MATHEMATICAL STATISTICS

MATH. 2611

SOLUTIONS TO THE FIRST MIDTERM EXAMINATION

SPRING, 2000

1.

  1. Beethoven wrote 9 symphonies and Mozart wrote 27 piano concertos. If a university radio station announcer wishes to play first a Beethoven symphony and then a Mozart concerto, in how many ways can this be done? 9x27=243
  2. The station manager decides that on each successive night, a Beethoven symphony will be played, followed by a Mozart piano concerto, followed by a Schubert string quartet (of which there are 15). For how many days could this policy be continued before exactly the same program would have to be repeated?

9x27x15=3645

2. Three molecules of type A, three molecules of type B, three molecules of type C, and three molecules of type D, are to be linked together to form a chain molecule.

  1. How many such chain molecules are there?
  2. In how many different ways all three molecules of each type end up next to one another (such as in BBBAAADDDCCC? 4x3x2x1=24

3. a. In a small town , 14 of the 25 school teachers are for the abortion. A random sample of five teachers is selected for an interview. In how many different ways 5 teachers can be selected so that all of them for abortion.

b. In how many different ways can seven persons form a circle for a folk dance? (7-1)!=6!=720

4. If A and B are mutually exclusive, P(A)=0.37, and P(B)=0.44, find

a. =0.37+0.44=0.81

b. =P(A)=0.37

c. =1-0.81=0.19

 

5. Suppose that P(A)=0.40 and P(B)=0.20. If the events A and B are independent, find these probabilities:

a. =0.4+0.20-(0.4)(0.2)=0.6-0.08=0.52

b. =P(A)P(B')=0.4(0.8)=0.32

6. Three missiles are fired at a target and hit it independently with probabilities 0.7, 0.8, and 0.9, respectively.

a. What is the probability that at least one of them will hit the target? 1-(0.3)(0.2)(0.1)=1-0.006=0.994

b. What is the probability that first one will miss, second one will hit and third one will miss the target? (0.3)(0.8)(0.1)=0.024

7. A fair die is thrown two times. Define the following events.

A={first toss is even}

B={second toss is even}

C={both tosses are the same}

a. Are A, B, and C independent events? Please show your work and justify your answer. P(A)=1/2, P(B)=1/2, P(C)=1/6, P(ABC)=P(22, 44, 66)=3/36=1/12 which is not same as P(A)P(B)P(C)=1/24. Therefore, they are not independent.

b. Are the events A and C mutually exclusive? A and C have common elements 22, 44, 66. Therefore they are not mutually exclusive.

HINT: The sample space for the experiment is

Second toss

First toss

1

2

3

4

5

6

1

(1,1)

(1,2)

(1,3)

(1,4)

(1,5)

(1,6)

2

(2,1)

(2,2)

(2,3)

(2,4)

(2,5)

(2,6)

3

(3,1)

(3,2)

(3,3)

(3,4)

(3,5)

(3,6)

4

(4,1)

(4,2)

(4,3)

(4,4)

(4,5)

(4,6)

5

(5,1)

(5,2)

(5,3)

(5,4)

(5,5)

(5,6)

6

(6,1)

(6,2)

(6,3)

(6,4)

(6,5)

(6,6)

8. An insurance company categorizes its customers as to whether they are high risk clients(H), medium risk clients(M), or low risk clients(L). They find that 10% are high risk, 30% are medium risk, and 60% are low risk. The company also noted that, over a given period of time, 5% of the high risk clients file a claim(C), 1% of the medium risk clients file a claim, and 0.1% of the low risk clients file a claim.

a. What is the probability that a customer will file a claim? P(C)=0.1(0.05)+0.6(0.01)+0.6(0.001)=0.005+0.003+0.0006=0.0086

b. If a customer files a claim, what is the probability that this person is a high risk client? P(H|C)=0.005/0.0086=0.5814.