Question

We choose a number from the set 1, 2, 3,..., 100 uniformly at random and denote this number by X. For each of the following choices decide whether the two events in question are independent or not.

a. A = {X is even), B-(X is divisible by 5}
b. C= (X has two digits), D = {X is divisible by 31}
c. E= {X is a prime), F = {X has a digit 5}

Answer 1

(a) A and B are dependent.

(b) C and D are dependent.

(c) E and F are dependent.

Explanation:

Two sets are said to be independent if the intersection of the two set is empty. That is the sets are disjoint.

Let's examine the situations to determine if they are independent or not.

(a) A = {2, 4, 6, . . . , 10, . . . , 20, . . .} and B = {5, 10, 15, . . . , 20, . . .}. Since A n B = {10, 20, . . .}, then the sets are dependent

(b) C = {10, 11, 12, . . . , 31, . . . , 62, . . . , 93, . . .} and D = {31, 62, 93}. Since C n D = {31, 62, 93}, then the sets are dependent.

(c) E = {5, 7, 11, . . .} and F = {5, 15, . . .}. Since E n F = {5}, then the set are dependent.

Thus

(a) A and B are dependent.

(b) C and D are dependent.

(c) E and F are dependent.