Are drawing a black and red card independent or mutually exclusive?

Are drawing a black card and queen independent?
Are drawing a face card and queen independent?

A black and red card are independent because you cannot draw a black card that is also a red card.

A black card and a queen are not independent because you can draw a card that is both black and a queen.

A face card and a queen are not independent because you can draw a card that is both a face card and a queen.

You are buying orange juice for $4.50 per container and have a gift card worth $7.00. The function f(x) = 4.50x - 7represents your total cost f(x) if you buy x containers of orange juice and use the gift card. How much do you pay to buy 4 containers of orange juice? How about if you buy 6?

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.