Sure! Let’s solve each part of the question step-by-step.
### Part (a): Probability that the card is a heart or a 7
First, let's determine the probabilities for each event separately and then find their combined probability using the principle of inclusion and exclusion.
1. Total number of cards in a standard deck: 52
2. Number of hearts in the deck:
- There are 13 hearts (one for each rank: Ace, 2, 3, ..., 10, Jack, Queen, King).
3. Number of 7s in the deck:
- There is one 7 for each suit (hearts, diamonds, clubs, spades), so there are 4 cards that are 7s.
4. Number of hearts that are also 7s:
- There is only one 7 of hearts.
Using the inclusion-exclusion principle, we find the combined probability:
1. Probability of drawing a heart:
[tex]\( P(\text{Heart}) = \frac{13}{52} \)[/tex]
2. Probability of drawing a 7:
[tex]\( P(\text{7}) = \frac{4}{52} \)[/tex]
3. Probability of drawing a card that is both a heart and a 7:
[tex]\( P(\text{Heart and 7}) = \frac{1}{52} \)[/tex]
Now, using the inclusion-exclusion principle:
[tex]\[ P(\text{Heart or 7}) = P(\text{Heart}) + P(\text{7}) - P(\text{Heart and 7}) \][/tex]
Plugging in the values:
[tex]\[ P(\text{Heart or 7}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{13 + 4 - 1}{52} = \frac{16}{52} = \frac{4}{13} \][/tex]
Calculate this to four decimal places:
[tex]\[ \frac{4}{13} \approx 0.3077 \][/tex]
So, the probability that the card is a heart or a 7 is 0.3077.
### Part (b): Probability that the first card is a heart and the second card is black (without replacement)
1. Total number of cards in the deck: 52
2. Number of hearts in the deck: 13
3. Number of black cards in the deck (spades and clubs): 26
For dependent events, we multiply the probabilities of each event occurring in sequence.
1. Probability of drawing a heart first:
[tex]\( P(\text{Heart first}) = \frac{13}{52} = \frac{1}{4} \)[/tex]
2. After drawing a heart, the number of remaining cards is 51. Of these, 26 are black.
3. Probability of drawing a black card second (after one heart is drawn):
[tex]\( P(\text{Black second}) = \frac{26}{51} \)[/tex]
The combined probability is:
[tex]\[ P(\text{Heart first, Black second}) = P(\text{Heart first}) \times P(\text{Black second}) \][/tex]
[tex]\[ P(\text{Heart first, Black second}) = \frac{13}{52} \times \frac{26}{51} \][/tex]
[tex]\[ P(\text{Heart first, Black second}) = \frac{13 \times 26}{52 \times 51} = \frac{338}{2652} \][/tex]
Simplify the fraction:
[tex]\[ \frac{338}{2652} = \frac{169}{1326} \approx 0.1274 \][/tex]
So, the probability that the first card is a heart and the second card is black (without replacement) is 0.1274.
### Summary
- Probability that the card is a heart or a 7: 0.3077
- Probability that the first card is a heart and the second card is black (without replacement): 0.1274
