Question
Question
Margarita draws two cards from a standard deck of cards. Event A
is to draw any card from the suit of hearts without replacing it and event B
is to draw any red card. What is P(A∩B)
? Note that there are 26 red cards in a standard deck and that a suit includes 13 cards. Round your answer to the nearest tenths place.(1 point)
Margarita draws two cards from a standard deck of cards. Event A
is to draw any card from the suit of hearts without replacing it and event B
is to draw any red card. What is P(A∩B)
? Note that there are 26 red cards in a standard deck and that a suit includes 13 cards. Round your answer to the nearest tenths place.(1 point)
Answers
GPT-4o mini
To find \( P(A \cap B) \), we first need to understand what events A and B represent:
- Event A: Drawing a card from the suit of hearts.
- Event B: Drawing any red card.
In a standard deck of cards:
- There are 26 red cards (13 hearts + 13 diamonds).
- There are 13 hearts (which are red cards).
When Margarita draws the first card:
- The probability of drawing a heart (Event A) is \( P(A) = \frac{13}{52} \).
If she draws a heart first and does not replace it, there will be 51 cards left in the deck, and she will have drawn one of the red cards.
Now for the intersection of events A and B (\( P(A \cap B) \)):
- If the first card is a heart (which is a red card), then the intersection consists of the case in which the first card drawn is a heart.
The probability of her drawing a heart first and it being a red card is simply the probability of her drawing a heart, as all hearts are red:
\[
P(A \cap B) = P(A) = \frac{13}{52}
\]
Calculating this gives us:
\[
P(A \cap B) = \frac{13}{52} = \frac{1}{4} = 0.25
\]
Since we need to round to the nearest tenths place, we write:
\[
P(A \cap B) \approx 0.3
\]
Thus, the final answer rounded to the nearest tenths place is:
\[
\boxed{0.3}
\]
- Event A: Drawing a card from the suit of hearts.
- Event B: Drawing any red card.
In a standard deck of cards:
- There are 26 red cards (13 hearts + 13 diamonds).
- There are 13 hearts (which are red cards).
When Margarita draws the first card:
- The probability of drawing a heart (Event A) is \( P(A) = \frac{13}{52} \).
If she draws a heart first and does not replace it, there will be 51 cards left in the deck, and she will have drawn one of the red cards.
Now for the intersection of events A and B (\( P(A \cap B) \)):
- If the first card is a heart (which is a red card), then the intersection consists of the case in which the first card drawn is a heart.
The probability of her drawing a heart first and it being a red card is simply the probability of her drawing a heart, as all hearts are red:
\[
P(A \cap B) = P(A) = \frac{13}{52}
\]
Calculating this gives us:
\[
P(A \cap B) = \frac{13}{52} = \frac{1}{4} = 0.25
\]
Since we need to round to the nearest tenths place, we write:
\[
P(A \cap B) \approx 0.3
\]
Thus, the final answer rounded to the nearest tenths place is:
\[
\boxed{0.3}
\]