Question 1185316
Here are examples of events A and B for a single card drawn from a standard 52-card deck, illustrating the four scenarios:

**(a) Mutually exclusive (disjoint) but not independent:**

*   **A:** The card drawn is a heart.
*   **B:** The card drawn is a spade.

These events are mutually exclusive because a card cannot be both a heart and a spade at the same time.  They are *not* independent because if you know the card is a heart (event A), you know it cannot be a spade (event B), and vice-versa.  The occurrence of one event affects the probability of the other.

**(b) Independent but not mutually exclusive:**

*   **A:** The card drawn is a face card (Jack, Queen, or King).
*   **B:** The card drawn is a club.

These events are independent because knowing the card is a face card does not change the probability that it is a club, and vice versa.  P(A) = 12/52, P(B) = 13/52, and P(A and B) = 3/52. Since P(A and B) = P(A)P(B), the events are independent. They are *not* mutually exclusive because there are three cards (Jack, Queen, and King of clubs) that are both face cards and clubs.

**(c) Independent and mutually exclusive:**

This is impossible with a single draw from a standard deck. If two events are mutually exclusive, then P(A and B) = 0. If two events are independent, then P(A and B) = P(A)P(B).  If both are true, then either P(A) or P(B) (or both) must be zero.  Since we are drawing a card, there is always at least one possible outcome for any event (e.g. the card could be an Ace of Spades, so the event "card drawn is a spade" has a non-zero probability).  Therefore, mutually exclusive events cannot be independent in this scenario (drawing a single card).

**(d) Neither independent nor mutually exclusive:**

*   **A:** The card drawn is a heart.
*   **B:** The card drawn is a red card.

These events are not mutually exclusive because a card can be both a heart and red. They are also not independent. If you know the card is a heart (event A), then you *know* it is red (event B). The occurrence of A completely determines B.  Therefore, they are dependent. P(A) = 13/52, P(B) = 26/52. P(A and B) = 13/52. P(A)P(B) = 13/52 * 26/52 = 676/2704 = 13/52. Since P(A and B) = P(A) it is not independent.