Question

1. Consider a fair four-sided die, with sides 1, 2, 3, and 4, that is rolled twice. For example, "1,4" would indicate 1 was rolled first and then 4 was rolled second a) Write down the possible outcomes, i.e., the sample space. (b) List the outcomes in the following events: Event A: The number 4 came up zero times. Event B: The number 4 came up exactly one time. . Event C: The sum of the two rolls is odd . Event D: The second roll was one more than the first roll. c) True or False: i. Events A and B are disjoint ii. Events B and C are disjoint iii. Event B is a subset of event C iv. Event D is a subset of event C. (d) Compute the probability P AUB (e) Compute the probability P[BUC]. (f) Compute the probability P CUD

Answer 1

a)

The exhaustative cases of two times throwing of a die is (Sample Space)

S = { (1,1),(1,2), (1,3), (1,4),

(2,1), (2,2), (2,3), (2,4),

(3,1), (3,2), (3,3), (3,4),

(4,1), (4,2), (4,3), (4,4), }
  
n = 16

b) i) A = { (1,1),(1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) }

ii) B = {(1,4), (2,4), (3,4), (4,1), (4,2), (4,3)}

iii) C = { (1,2), (1,4), (2,1), (2,3), (3,2), (3,4), (4,1), (4,3) }

iv) D = { (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) }

c) i) True

ii) False

iii) False

iv) False

d) P(AUB) = P(A) + P(B) = 9/16 + 6/16 = 14/16

e) P(BUC) = P(B) + P(C) - P(B and C) = 6/16 + 8/16 - 3/16 = 11/16

f)  P(CUD) = P(C) + P(D) - P(C and D) = 8/16 + 6/16 - 4/16 = 10/16