d) Are the events King and face card independent events? Why or why not? Round answer to 3 decimal places. Select an answer

Question

UpStudy AI · Accepted Answer

Let

- $$ K $$ be the event that a card is a King.
- $$ F $$ be the event that a card is a face card (Jack, Queen, or King).

There are 52 cards in a standard deck.

1. The probability of drawing a King is
   $$
   P(K) = \frac{4}{52} \approx 0.077.
   $$

2. The probability of drawing a face card is
   $$
   P(F) = \frac{12}{52} \approx 0.231.
   $$

3. Since every King is a face card, the probability of drawing a card that is both a King and a face card is
   $$
   P(K \cap F) = \frac{4}{52} \approx 0.077.
   $$

4. If the events were independent, then
   $$
   P(K \cap F) = P(K) \times P(F) \approx 0.077 \times 0.231 \approx 0.018.
   $$
   However, we found that
   $$
   P(K \cap F) \approx 0.077 \neq 0.018.
   $$

Because $$P(K \cap F) \neq P(K) \times P(F)$$, the events $$K$$ and $$F$$ are not independent.

Final answer: They are not independent.
They are not independent because the probability of both events occurring together is not equal to the product of their individual probabilities.

d) Are the events King and face card independent events? Why or why not? Round answer to 3 decimal places. Select an answer

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