Download Midterm 2 Practice, Part C

Midterm 2 Practice, Part C
The following problems have been given at ¯nal exams of previous years or they are practice
problems that we wrote and are are relatively close to questions you may see at the exam. Recall
that the ¯nal exam will also contain material that was covered in Midterm 1 and Midterm 2.
Spend some time with the midterm practice sets as well. We have also included few problems that
correspond to material of Midterm 1 and Midterm 2.
Problem 1 This is a multiple choice question. In each part there are three choices, 1, 2 or 3. You
must circle the number corresponding to the best choice in each case. Points are awarded like this:
² A correct answer on any part is worth four points,
² Leaving it blank gets you zero points, but
² For a wrong answer, 2 points will be deducted, meaning there is a penalty for guessing.
(1)
(2)
The recurrence an = 2 + an¡1 (n ¸ 1) with a0 = 2 speci¯es the same sequence with the
explicit formula
1.
an = 2n for n ¸ 0
2.
an = 2(n + 1) for n ¸ 0
3.
both of the above
Suppose A; B are events such that P (A \ B) = 0:2 and P (B) = 0:8. Then P (AjB)
1.
is 0:25
2.
is 0:16
3.
cannot be determined from the given information
Problem 2
1. P (AjB) =
(a) P (A) + P (B) ¡ P (A \ B)
(b) P (A) ¡ P (B) + P (A \ B)
(c) P (A \ B)=P (B)
2. P (A1 \ A2 \ A3 ) = : : :
(a) P (A1 jA2 \ A3 )P (A2 jA3 )P (A3 )
(b) P (A1 )P (A2 )P (A3 )
1
(c) none of the above
3. The random variable X follows the normal distribution Á¹;¾ . Which of the following hold?
(a) P (X > ¹) =
1
2
(b) P (X > ¹jX > ¡¹) =
1
4
(c) both of the above
4. We toss a fair coin 100 times. The probability to get heads 50 times is
(a) 50=100
(100)
(b) 25050
(c) none of the above
Problem 3 Consider two boxes A and B ¯lled with red and blue balls. Box A has 8 red and 16
blue balls. Box B has 6 red and 12 blue balls. First you choose one of the two boxes. Box A has
probability 0.4 to be chosen and box B has probability 0.6 to be chosen. Let us assume that you
choose at random 6 balls out of the chosen box, sampling without replacement.
What is the probability that you have picked 4 red and 2 blue balls?
Given that you have picked 4 red and 2 blue balls, what is the probability that you picked them
from box A?
Again, you pick 6 balls from the chosen box. But now if the red balls are more than the blue
balls then you pick one more ball from the other box. Let us call B the random variable
that represents the total number of blue balls that you end up with. Compute its probability
distribution.
Problem 4 Suppose a sequence satis¯es the given recurrence relation and initial conditions. Find
an explicit formula for each sequence.
1.
an = ak¡1 + 2ak¡2
a0 = 6; a1 = 3
2.
bn = 6bk¡1 ¡ 9bk¡2
b0 = 0; b1 = 3
Problem 5 A sequence is de¯ned recursively as follows:
vk = 2vk¡2 ; 8k ¸ 2;
v0 = 1; v1 = 2
1. Use iteration to guess an explicit closed formula (a function of n) for the sequence.
2
2. Use induction to prove that the formula in part (a) holds for all n.
Problem 6 Suppose you have 2 coins in your pocket, one fair coin (equal chance of H or T) and
one fake coin with 23 chance of H and 13 chance of T. Suppose you randomly pick one coin from
your pocket.
1. What is the probability the coin is fair and when you °ip it you get tails.
2. What is the probability the coin is fake and when you °ip it you get heads.
3. Suppose you °ip the coin and heads comes up, what is the probability the coin you °ipped
is the fair one? The fake one?
Problem 7 Box A contains 6 balls of which 1 is defective and box B contains 7 balls of which 3
are defective. A ball is drawn at random from each box. If one ball is defective and one is not,
what is the probability that the defective one came from box A?
Problem 8 CSE999 is taken by both CSE, Physics, and Math majors. The grade Xc of a random
CSE major follows the normal distribution Á¹c ;¾ . The grade Xp of a random Physics major follows
the normal distribution Á¹p ;¾ and the grade of a random Math major is Á¹m ;¾ . 40 CSE students,
20 Math Students and 20 Physics students take the class.
Finally, assume that ¹p < ¹c < ¹m and in addition it is ¹c = ¹p + ² and ¹m = ¹c + ².
Assume that ¹c = 50, ¹p = 40, ¹m = 60, and ¾ = 10. With the help of the attached standard
normal curve area table ¯nd the following:
1. The probability P (Xc > 45).
2. The probability P (Xp > 45).
3. The probability P (Xm < 55).
4. The probability P (Xc < 55).
5. The probability P (45 < Xc < 55).
6. The probability P (45 < Xm < 55).
7. Find the probability that the grade of a random student is greater than ¹c . Do not use the
speci¯c numbers for ¹c , ¹p , ¹m and ¾.
Problem 9 A family has 6 children.
1. Assuming that each kid may be a boy with probability
family has exactly 3 boys.
1
2
what is the probability that the
2. It turns out that according to recent detailed statistics the probability of a kid being a boy
is 0:498. What is the probability that the family has exactly 3 boys?
Problem 10 There are two boxes, A and B. The ¯rst box contains 7 blue and 3 yellow marbles.
The second box contains 5 blue and 5 yellow marbles. 4 marbles are selected randomly from a
random box.
3
1. What is the probability that 3 blue and 1 yellow marble are selected?
2. Assuming that 3 blue and 1 yellow marble are selected, what is the probability that the box
A was selected.
Problem 11 Assume that you have an in¯nitly large set of blocks of heights 1 and 2 inches.
Imagine constructing towers by piling blocks of di®erent heights directly on top of one another. For
example, a tower of height 6 inches could be obtained using any of the following sequences:
² Six 1-inch blocks. This sequence is represented as [1; 1; 1; 1; 1; 1].
² Three 2-inch blocks. This sequence is represented as [2; 2; 2].
² One 1-inch block, stacked on top of one 2-inch block, stacked on top of one 1-inch block,
stacked on top of one 2-inch block. This sequence is represented as [2; 1; 2; 1]. Note that the
sequence [1; 2; 1; 2] is di®erent from the sequence [2; 1; 2; 1] or the sequence [1; 1; 2; 2]. That
is, the order is important.
² etc...
Let tn be the number of ways to construct a tower of height n inches using blocks from the set.
1. Find a recurrence relation for tn .
2. Solve the recurrence relation tn , i.e., derive an explicit formula for tn .
4