To determine whether drawing a black and red card is independent or mutually exclusive, we need to consider whether one event affects the probability of the other event occurring.
If we have a standard deck of playing cards, it contains 26 black cards (clubs and spades) and 26 red cards (hearts and diamonds). The events of drawing a black card and drawing a red card are mutually exclusive because a card cannot be both black and red at the same time. Therefore, if you draw a black card, the probability of drawing a red card is zero, and vice versa. These events are not independent.
Now let's consider drawing a black card and a queen. In a standard deck, there are 26 black cards, and there are 4 queens, of which 2 are black (the queen of clubs and the queen of spades). In this case, the events of drawing a black card and drawing a queen are not mutually exclusive because a black card can also be a queen. However, these events are still not independent because drawing a black card affects the probability of drawing a queen. If you draw a black card, the probability of drawing a queen increases, as there are more black queens in the deck compared to non-black queens.
Finally, let's look at drawing a face card and a queen. In a standard deck, there are 12 face cards (4 jacks, 4 queens, and 4 kings), of which 4 are queens. Similarly to the previous scenario, the events of drawing a face card and drawing a queen are not mutually exclusive because a queen is considered a face card. However, these events are also not independent because drawing a face card affects the probability of drawing a queen. If you draw a face card, the probability of drawing a queen decreases, as only one-fourth of the face cards are queens.
In summary:
- Drawing a black and red card: Mutually exclusive and not independent.
- Drawing a black card and queen: Not mutually exclusive and not independent.
- Drawing a face card and queen: Not mutually exclusive and not independent.