Question 1185316: A single card is drawn from an ordinary deck. Give examples of events A and B
associated with this experiment that are
(a) Mutually exclusive (disjoint) but not independent
(b) Independent but not mutually exclusive
(c) Independent and mutually exclusive
(d) Neither independent nor mutually exclusive
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! 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.
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