We choose a number from the set {1, 2, 3, . . . , 100}, uniformly at random. Let X be the chosen number. For each of the following choices determine if the two events are independent or not. Explain why. (a) A = {X is even), B = (X is divisible by 5}
(b) C = (X has two digits), D = {X is divisible by 3}
(c) E = {X, is a prime}, F = has a digit 5 prime number

(a) A and B are not independent.

(B) C and D are not independent.

(C) E and F are not independent.

Step-by-step explanation:

Two events are called independent If the output sets for the two events do not have any common element.

i.e. their intersection is Null.

If P and Q are independent events then

Now, let us have a look at the given questions and find whether they are independent or not as per above definition

(a) A = {X is even}, B = {X is divisible by 5}

A = {2, 4, 6, 8, 10, .....20, 22, 24, ..... }

B = {5, 10, 15, 20,  ....}

A and B have common elements {10, 20, 30, ....}

So, the two events are not independent.

(b) C = {X has two digits}, D = {X is divisible by 3}

C = {11, 12, 13, ......, 99}

D = {3, 6, 9 12, 15, .... , 99}

C and D have common elements {3, 6, 9 12, 15, .... , 99}

So, the two events are not independent.

(c) E = {X, is a prime}, F = has a digit 5 prime number

E = {2, 3, 5, 7, 11, 13, 17, 19, 23, .....}

F = {5}

E and F have common element {5}

So, the two events are not independent.

So, the answer are:

(a) A and B are not independent.

(B) C and D are not independent.

(C) E and F are not independent.