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)
percent
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)
percent
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. There are 13 hearts in the deck.
- Event B: Drawing any red card. There are 26 red cards in total (13 hearts and 13 diamonds).
We are interested in the intersection of events A and B, denoted by \( A \cap B \), which represents drawing a card that is both from the suit of hearts and a red card.
Since all hearts are red cards, the cards that satisfy both events A and B are the 13 hearts.
Now, we calculate \( P(A \cap B) \):
1. **Calculate the probability of drawing a heart and it being a red card:**
Since all hearts are red, when you draw the first card, the probability of drawing a heart (which is also a red card) is:
\[
P(A \cap B) = P(A) = \frac{13}{52} = \frac{1}{4}
\]
2. **Since this is the first card drawn, we should check the impact on the second card:**
For two draws without replacing the first card, however, if you draw a heart (which is a red card) first, you now have 51 cards left in the deck.
Assuming the first card drawn was a heart (a red card), in the second scenario, there are still 12 hearts left and 25 remaining red cards (13 diamonds and now 12 hearts since one heart has already been drawn).
3. **Calculate the total probabilities involving two draws:**
In the first draw, the heart drawn contributes to the \( A \) and thus \( A \cap B \) situation.
Since we only need the probability of the first draw:
- P(A and red) is just the probability of drawing a heart initially:
\[
P(A \cap B) = \frac{13}{52}
\]
To convert this probability to a percentage, you multiply by 100:
\[
P(A \cap B) = \left(\frac{13}{52}\right) \times 100 = 25\%
\]
Thus, rounding to the nearest tenths, the answer is:
**25.0%**
- Event A: Drawing a card from the suit of hearts. There are 13 hearts in the deck.
- Event B: Drawing any red card. There are 26 red cards in total (13 hearts and 13 diamonds).
We are interested in the intersection of events A and B, denoted by \( A \cap B \), which represents drawing a card that is both from the suit of hearts and a red card.
Since all hearts are red cards, the cards that satisfy both events A and B are the 13 hearts.
Now, we calculate \( P(A \cap B) \):
1. **Calculate the probability of drawing a heart and it being a red card:**
Since all hearts are red, when you draw the first card, the probability of drawing a heart (which is also a red card) is:
\[
P(A \cap B) = P(A) = \frac{13}{52} = \frac{1}{4}
\]
2. **Since this is the first card drawn, we should check the impact on the second card:**
For two draws without replacing the first card, however, if you draw a heart (which is a red card) first, you now have 51 cards left in the deck.
Assuming the first card drawn was a heart (a red card), in the second scenario, there are still 12 hearts left and 25 remaining red cards (13 diamonds and now 12 hearts since one heart has already been drawn).
3. **Calculate the total probabilities involving two draws:**
In the first draw, the heart drawn contributes to the \( A \) and thus \( A \cap B \) situation.
Since we only need the probability of the first draw:
- P(A and red) is just the probability of drawing a heart initially:
\[
P(A \cap B) = \frac{13}{52}
\]
To convert this probability to a percentage, you multiply by 100:
\[
P(A \cap B) = \left(\frac{13}{52}\right) \times 100 = 25\%
\]
Thus, rounding to the nearest tenths, the answer is:
**25.0%**