Can someone please help me. A single card is selected from a standard 52-card deck. B= the drawn card is black; R= the drawn card is red; Q= the drawn card is a queen; F= the drawn card is a face card ( a king, queen or jack). Without finding the probabilities, determine if B and R are independent, without finding probabilities, determine if B and Q are independent, without finding the probabilities, determine if F and Q are independent. Using the probabilities, determine if B and R are independent. Using the probabilities determine if B and Q are independent. Using the probabilities, determine if F and Q are independent.

To determine if two events are independent, we need to compare the probability of their joint occurrence to the product of their individual probabilities. If the joint probability is equal to the product of the individual probabilities, then the events are independent.

Without finding the probabilities:
1. To determine if B (the drawn card is black) and R (the drawn card is red) are independent, we can conclude that they are dependent. This is because if the drawn card is black, then it cannot be red, and vice versa. Therefore, B and R are mutually exclusive events and cannot occur together.

2. To determine if B (the drawn card is black) and Q (the drawn card is a queen) are independent without finding probabilities, we cannot draw any immediate conclusion. We need to find the probabilities to make a determination.

3. To determine if F (the drawn card is a face card) and Q (the drawn card is a queen) are independent without finding probabilities, we cannot draw any immediate conclusion. We need to find the probabilities to make a determination.

Using the probabilities:
To definitively determine if two events are independent, we must calculate their respective probabilities and compare them.

1. B and R:
We need to find P(B), P(R), and P(B ∩ R) to determine independence. The probability of drawing a black card is 26/52 = 1/2, as there are 26 black cards out of a total of 52. Similarly, the probability of drawing a red card is also 1/2. However, the probability of drawing both a black card and a red card simultaneously is 0, as it is impossible to draw a card that is both black and red. Since P(B ∩ R) ≠ P(B) * P(R), we can conclude that B and R are dependent events.

2. B and Q:
Again, we need to find P(B), P(Q), and P(B ∩ Q) to determine independence. To find P(B ∩ Q), we need to determine the number of black queens in a deck. There are 2 black queens in a standard deck, so P(B ∩ Q) = 2/52 = 1/26. The probability of drawing a black card (P(B)) and the probability of drawing a queen (P(Q)) can be calculated as 1/2 and 4/52 respectively. Since P(B ∩ Q) = P(B) * P(Q), we can conclude that B and Q are independent events.

3. F and Q:
Similarly, we need to find P(F), P(Q), and P(F ∩ Q) to determine independence. The probability of drawing a face card (F) is 12/52, as there are 12 face cards in a deck. The probability of drawing a queen (Q) is 4/52. To find P(F ∩ Q), we need to determine the number of face cards that are also queens, which is 4. Therefore, P(F ∩ Q) = 4/52. Since P(F ∩ Q) ≠ P(F) * P(Q), we can conclude that F and Q are dependent events.