Question

A deck of cards is shuffled and a card is drawn. Let A be the event that the card is a heart and B be the event that the card is a Jack.

(a) Calculate the following probabilities: P(A),P(B),P(A∩B),P(A∪B),P(A∣B),P(B∣A).
(b) Are the events A and B independent? Explain briefly.
(c) Are the events A and B mutually exclusive? Explain briefly.

Answer 1

To calculate the probabilities, we use the number of hearts and Jacks in a standard deck of cards. P(A) = 1/4, P(B) = 1/13, P(A∩B) = 1/52, P(A∪B) = 4/13, P(A|B) = 1/13, P(B|A) = 1/4. Events A and B are not independent but are mutually exclusive.

Step-by-step explanation:

To calculate the probabilities, we need to know the number of hearts and Jacks in a standard deck of cards.

There are 13 hearts and 4 Jacks in a deck of 52 cards.

(a) P(A) = 13/52 = 1/4, P(B) = 4/52 = 1/13, P(A∩B) = 1/52, P(A∪B) = P(A) + P(B) - P(A∩B) = 16/52 = 4/13, P(A|B) = P(A∩B) / P(B) = 1/13, P(B|A) = P(A∩B) / P(A) = 1/4.

(b) Events A and B are independent if P(A∩B) = P(A) * P(B), which is not the case since P(A∩B) = 1/52 but P(A) * P(B) = (1/4) * (1/13) = 1/52.

(c) Events A and B are mutually exclusive if P(A∩B) = 0, which is the case here since P(A∩B) = 1/52 ≠ 0.