Question

Consider two urns of balls: the first contains 5 different red balls numbered from 1 to 5 and the second contains 4 different blue balls numbered from 1 to 4. You are asked to pick one ball from the first urn (i.e., the one with red balls) and one ball from the second urn (i.e., the one with blue balls). Each outcome has the form (r, b), where r denotes the number on the red ball and b denotes the number on the blue ball. For example (2, 3) corresponds to picking the red ball with number 2 and the blue ball with number 3. (a) Write the sample space S. (b) Write the set representing the event A1 = {b is either 1 or a prime number}. 1 (c) Write the set representing the event A2 = {r is an even number and b is an odd number}. (d) Write the set representing the event A3 = {the product of r and b is an even number}. (e) Write the set representing the event B1 = {the sum of r and b is greater than or equal to 5}. (f) Write the set representing the event B2 = {r and b are equal}. (g) Write the set representing the event B3 = {r is an odd number and b is either 1 or 3}. For each pair of sets A1 and B1, A2 and B2, A3 and B3, identify whether the pair of sets is either mutually exclusive or collectively exhaustive or both.

Answer 1

(a) 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),(5,1),(5,2),(5,3),(5,4)}.
n(S) = 20.

(b) A1 = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(4,1),(4,2),(4,3),(5,1),(5,2),(5,3)}.
n(A1) = 15.

(c) A2 = {(2,1),(2,3),(4,1),(4,3)}.
n(A2) = 4.

(d) A3 = {(1,2),(1,4),(2,1),(2,2),(2,3),(2,4),(3,2),(3,4),(4,1),(4,2),(4,3),(4,4),(5,2),(5,4)}.
n(A3) = 14.

(e) B1 = {(1,4),(2,3),(2,4),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4),(5,1),(5,2),(5,3),(5,4)}.
n(B1) = 14.

(f) B2 = {(1,1),(2,2),(3,3),(4,4)}.
n(B2) = 4.

(g) B3 = {(1,1),(1,3),(3,1),(3,3),(5,1),(5,3)}.
n(B3) = 6.

(h) For sets A1 and B1:
A1 and B1 can occur simultaneously in some cases. Hence, they are not mutually exclusive. They are collectively exhaustive, though, as they cover the entire sample space.
For sets A2 and B2:
A2 and B2 are mutually exclusive as they cannot occur simultaneously. They are not collectively exhaustive, though, as they do not cover the entire sample space.
For sets A3 and B3:
A3 and B3 are mutually exclusive as they cannot occur simultaneously. They are also collectively exhaustive because they cover the entire sample space.