Question

Two cards are chosen at random from a standard 52-card deck. Consider the events: A = first card is the ace of spades B = second card is the ace of spades. Suppose the two cards are selected with replacement.

i. Are the events A and B independent? Why?

ii. Are the events A and B mutually exclusive? Why?

Now suppose the two cards are selected without replacement.

iii. Are the events A and B mutually exclusive? Why?

iv. Are the events A and B independent? Why?

Answer 1

(i) Yes, Since cards are drawn with replacement, So probability of ace of spade on first draw & on second draw will not be affected by each other. P(B) will be 1/52 irrespective of result of event A.

(ii) No, Since cards are drawn with replacement, so it is possible to get ace of spade on both first and second draw. so, A and B both can occur simultaneously. Hence, they are not mutually exclusive.

(iii) Yes, Since cards are drawn without replacement, so if first card is ace of spade, second card cant be ace of spade and vice versa, so A and B both cant occur simultaneously. Hence they are mutually exclusive.

(iv) No, Since cards are drawn without replacement, so if first card is ace of spade, second card cant be ace of spade, but if first card is not ace of spade, second card can be ace of spade. This means probability of occurance of event B depends on outcomes of event A.