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, you need to compare their probabilities. If the probability of one event occurring does not change based on whether the other event has occurred or not, then the events are considered independent.
1. Without finding probabilities, determine if B and R are independent:
Since black cards and red cards are mutually exclusive in a standard 52-card deck, meaning a card cannot be both black and red, the occurrence of one event (B or R) always determines the other event. Thus, B and R are dependent.
2. Without finding probabilities, determine if B and Q are independent:
Since there are 2 black queens and 2 red queens in a standard 52-card deck, the occurrence of one event (B or Q) influences the probability of the other event. Therefore, B and Q are dependent.
3. Without finding probabilities, determine if F and Q are independent:
Again, without considering probabilities, we know that there are 12 face cards in a standard deck (4 kings, 4 queens, and 4 jacks) and 4 queens. The occurrence of one event (F or Q) affects the probability of the other event. Thus, F and Q are dependent.
Now, let's determine whether these events are independent using probabilities:
4. Using the probabilities, determine if B and R are independent:
To determine independence, we need to calculate the probabilities:
P(B) = Number of black cards / Total number of cards = 26/52 = 1/2
P(R) = Number of red cards / Total number of cards = 26/52 = 1/2
If these events were independent, the joint probability would be equal to the product of the individual probabilities:
P(B ∩ R) = P(B) * P(R)
P(B ∩ R) = (1/2) * (1/2) = 1/4
However, since P(B ∩ R) is not equal to 1/4, B and R are dependent.
5. Using the probabilities, determine if B and Q are independent:
To determine independence, we need to calculate the probabilities:
P(B) = 26/52 = 1/2
P(Q) = 4/52 = 1/13
If these events were independent, the joint probability would be equal to the product of the individual probabilities:
P(B ∩ Q) = P(B) * P(Q)
P(B ∩ Q) = (1/2) * (1/13) = 1/26
Since P(B ∩ Q) is not equal to 1/26, B and Q are dependent.
6. Using the probabilities, determine if F and Q are independent:
To determine independence, we need to calculate the probabilities:
P(F) = 12/52 = 3/13
P(Q) = 4/52 = 1/13
If these events were independent, the joint probability would be equal to the product of the individual probabilities:
P(F ∩ Q) = P(F) * P(Q)
P(F ∩ Q) = (3/13) * (1/13) = 3/169
Since P(F ∩ Q) is not equal to 3/169, F and Q are dependent.
In conclusion:
- Without finding probabilities, we determined that B and R, B and Q, and F and Q are all dependent.
- Using the probabilities, we confirmed that B and R, B and Q, and F and Q are indeed dependent.