## Description

Exercise 1.

Consider an experiment consisting of rolling a die twice. The outcome of this experiment is an ordered pair

whose first element is the first value rolled and whose second element is the second value rolled.

(a) Find the sample space.

(b) Find the set A representing the event that the value on the first roll is greater than or equal to the

value on the second roll.

(c) Find the set B corresponding to the event that the first roll is a six.

(d) Let C correspond to the event that the first valued rolled and the second value rolled differ by two.

Find A ∩ C.

Note that A, B, and C should be subsets of the sample space specified in Part (a).

c 2020 Stanley Chan. All Rights Reserved. 1

Exercise 2.

A space S is defined as S = {1, 3, 5, 7, 9, 11}, and three subsets as A = {1, 3, 5}, B = {7, 9, 11}, C =

{1, 3, 9, 11}. Assume that each element has probability 1/6. Find the following probabilities: (a) P[A], (b)

P[B], (c) P[C], (d) P[A ∪ B], (e) P[A ∪ C], (f) P[(A\C) ∪ B].

Exercise 3.

Let the events A and B have P[A] = x, P[B] = xy and P[A ∪ B] = x

2

z. Find the following probabilities.

(a) P[A ∩ B]

(b) P[Ac ∩ Bc

]

(c) P[Ac ∪ Bc

]

(d) P[A ∩ Bc

]

(e) P[Ac ∪ B]

c 2020 Stanley Chan. All Rights Reserved. 2

Exercise 4.

A number x is selected at random in the interval [−1, 3]. Let the events A = {x | x < 0}, B = {x |(x − 1)2 <

4}, C = {x | x > 1}. Find (a) P[A | B], (b) P[B | C], (c) P[A | C

c

], (d) P[B | C

c

].

Exercise 5.

Let A, B, C be events with probabilities P[A] = 0.3, P[B] = 0.2, P[C] = 0.5. Find

(a) P[A ∪ B] if A and B are independent

(b) P[A ∪ B] if A and B are disjoint

(c) P[A ∪ B ∪ C] if A, B and C are independent

(d) P[A ∪ B ∪ C] if A, B and C are pairwise disjoint; Can this happen?

c 2020 Stanley Chan. All Rights Reserved. 3

Exercise 6.

(a) By using the fact that P[A ∪ B] ≤ P[A] + P[B], show that P[A ∪ B ∪ C] ≤ P[A] + P[B] + P[C].

(b) By using the fact that P [

Sn

k=1 Ak] ≤

Pn

k=1 P[Ak], show that P [

Tn

k=1 Ak] ≥ 1 −

Pn

k=1 P[Ac

k

].

c 2020 Stanley Chan. All Rights Reserved. 4

Exercise 7.

The following result is known as the Bonferroni’s Inequality.

(a) Prove that for any two events A and B, we have

P(A ∩ B) ≥ P(A) + P(B) − 1.

(b) Generalize the above to the case of n events A1, A2, . . . , An, by showing that

P(A1 ∩ A2 ∩ . . . ∩ An) ≥ P(A1) + P(A2) + . . . + P(An) − (n − 1).

Hint: You may use the generalized Union Bound P(

Sn

i=1 Ai) ≤

Pn

i=1 P(Ai).

c 2020 Stanley Chan. All Rights Reserved. 5

Exercise 8.

A block of information is transmitted repeated over a noisy channel until an error-free block is received. Let

M ≥ 1 be the number of blocks required for a transmission. Define the following sets

(i) A = {M is even}

(ii) B = {M is a multiple of 5}

(iii) C = {M is less than or equal to 7}

Assume that the probability of requiring one additional block is half of the probability without the additional

block. That is:

P[M = k] =

1

2

k

, k = 1, 2, . . . .

Determine the following probabilities

(a) P[A], P[B], P[C], P[C

c

]

(b) P[A ∩ B], P[A\B], P[A ∩ B ∩ C]

(c) P[A | B], P[B | A]

(d) P[A | B ∩ C], P[A ∩ B | C]

Hint: P[A] 6=

1

2

. P[A] = P

k=even

P[M = k].

c 2020 Stanley Chan. All Rights Reserved. 6

c 2020 Stanley Chan. All Rights Reserved. 7

Exercise 9. (Programming)

Write a MATLAB / Python program to simulate the following experiments.

(a) Draw a dice 100 times. That is, generate a sequence of 100 random numbers from the set {1, . . . , 6}.

Call this sequence X1, . . . , X100. Plot the histogram of X1, . . . , X100, with bin centers {1, . . . , 6}. Do

not use a for-loop in your code. Submit your plot.

(b) Repeat (a) by drawing the dice 10000 times.

(c) Draw another dice 100 times. Call this sequence Y1, . . . , Y100. Let Zi = Xi +Yi for i = 1, . . . , 100. Plot

the histogram of Z1, . . . , Z100. Submit your plot.

(d) Repeat (c) by drawing the dices 10000 times.

(e) Using the histogram found in (d), find the probability that 4 < Zi ≤ 7.

(f) In (c)-(d), Zi

is a sum of two random variables Xi and Yi

. What if we sum more random variables?

That is, Zi = X

(1)

i + X

(2)

i + . . . + X

(K)

i

. Let K = 10. Plot the histogram of {Z1, . . . , Z10000}. Submit

your histogram. Pay attention to the bin centers.

(g) Repeat (f) by setting K = 100. Plot the histogram of {Z1, . . . , Z10000}. Submit your histogram. Pay

attention to the bin centers.

Please put your plots after this page.

c 2020 Stanley Chan. All Rights Reserved. 8