Math 230 Final exam Fall 1998 Solutions

1. Let T = the set of students who do have a big screen TV
Let NT = the set of students who do not have a big screen TV
Then

2. Since A and B are independent . Also, then

Since , . By drawing the Venn diagram, it can be seen that , so
= Pr(C| A)Pr(A) + Pr(C|B)Pr(B) .

So AND

3.

4. 12 eggs = 10 good + 2 rotten. When you pick 4 eggs, you can pick all good eggs, 1 bad and 3 good, or 2 good and 2 bad. There are no other choices because there are only 2 bad eggs. Let X = the number of rotten eggs. So the possible values of the random variable X are 0, 1, and 2.

Also,

5. n = 1,000,000, ,
a)
b)
c)
d)

6. (a) Using matrix inverse. If then find

So

Since

So x = 4 and y = -2.

(b) (using augmented matrix)

So, x = 2 and y = 4.

7. (i) Transition diagram: (Four states)

Transition matrix:

(ii)

So and

= =

(iii) (BONUS)
So the probability of getting two 3's (state 2) is , since the starting state is state 4.

8. (a) Since each row of the transition matrix of a Markov chain must add to 1, so and so .

(b) Let .

We want to find so that

We get and . Also, we know that .

Solving for and using matrices

So and

9.
x + y + s = 20
2x + y + t = 20
(slack)

maximize y-x f + x - y = 0

No more negatives, STOP! So, y=20, x=0, f=20The maximum value of y - x is 20.

10.
x +y = 15 x + y + a = 20
y -2x + s = 0
( a surplus, s slack)

maximize y y - Ma f - y + Ma= 0
Let M = 100

No more negatives, so STOP! x = 5, y= 10 and f = 10. So the maximum value of y is 10.