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Sagot :
Let's solve this step-by-step:
### a) Probability of drawing either a black card or a face card
Given:
- Total number of cards = 52
- Number of black cards (spades and clubs) = 26
- Number of face cards in the deck (jack, queen, king of each suit) = 12
- Number of black face cards (face cards in spades and clubs) = 6
Using the principle of inclusion-exclusion:
[tex]\[ P(\text{Black or Face}) = \frac{(\text{Number of Black Cards} + \text{Number of Face Cards} - \text{Number of Black Face Cards})}{\text{Total Number of Cards}} = \frac{(26 + 12 - 6)}{52} = \frac{32}{52} = 0.6153846153846154 \][/tex]
So, the probability is [tex]\(0.615 \approx 0.62\)[/tex].
### b) Probability of drawing either a red card or an ace
Given:
- Total number of cards = 52
- Number of red cards (hearts and diamonds) = 26
- Number of aces in the deck = 4
- Number of red aces (aces in hearts and diamonds) = 2
Using the principle of inclusion-exclusion:
[tex]\[ P(\text{Red or Ace}) = \frac{(\text{Number of Red Cards} + \text{Number of Aces} - \text{Number of Red Aces})}{\text{Total Number of Cards}} = \frac{(26 + 4 - 2)}{52} = \frac{28}{52} = 0.5384615384615384 \][/tex]
So, the probability is [tex]\(0.538 \approx 0.54\)[/tex].
### c) Probability of drawing either a spade or a queen card
Given:
- Total number of cards = 52
- Number of spades = 13
- Number of queens in the deck = 4
- Number of spade queens = 1 (only the queen of spades)
Using the principle of inclusion-exclusion:
[tex]\[ P(\text{Spade or Queen}) = \frac{(\text{Number of Spades} + \text{Number of Queens} - \text{Number of Spade Queens})}{\text{Total Number of Cards}} = \frac{(13 + 4 - 1)}{52} = \frac{16}{52} = 0.3076923076923077 \][/tex]
So, the probability is [tex]\(0.308 \approx 0.31\)[/tex].
### d) Probability of drawing either a heart or a jack
Given:
- Total number of cards = 52
- Number of hearts = 13
- Number of jacks in the deck = 4
- Number of heart jacks = 1 (only the jack of hearts)
Using the principle of inclusion-exclusion:
[tex]\[ P(\text{Heart or Jack}) = \frac{(\text{Number of Hearts} + \text{Number of Jacks} - \text{Number of Heart Jacks})}{\text{Total Number of Cards}} = \frac{(13 + 4 - 1)}{52} = \frac{16}{52} = 0.3076923076923077 \][/tex]
So, the probability is [tex]\(0.308 \approx 0.31\)[/tex].
These are the probabilities for each of the given events:
1. Probability of drawing either a black card or a face card: 0.615
2. Probability of drawing either a red card or an ace: 0.538
3. Probability of drawing either a spade or a queen: 0.308
4. Probability of drawing either a heart or a jack: 0.308
### a) Probability of drawing either a black card or a face card
Given:
- Total number of cards = 52
- Number of black cards (spades and clubs) = 26
- Number of face cards in the deck (jack, queen, king of each suit) = 12
- Number of black face cards (face cards in spades and clubs) = 6
Using the principle of inclusion-exclusion:
[tex]\[ P(\text{Black or Face}) = \frac{(\text{Number of Black Cards} + \text{Number of Face Cards} - \text{Number of Black Face Cards})}{\text{Total Number of Cards}} = \frac{(26 + 12 - 6)}{52} = \frac{32}{52} = 0.6153846153846154 \][/tex]
So, the probability is [tex]\(0.615 \approx 0.62\)[/tex].
### b) Probability of drawing either a red card or an ace
Given:
- Total number of cards = 52
- Number of red cards (hearts and diamonds) = 26
- Number of aces in the deck = 4
- Number of red aces (aces in hearts and diamonds) = 2
Using the principle of inclusion-exclusion:
[tex]\[ P(\text{Red or Ace}) = \frac{(\text{Number of Red Cards} + \text{Number of Aces} - \text{Number of Red Aces})}{\text{Total Number of Cards}} = \frac{(26 + 4 - 2)}{52} = \frac{28}{52} = 0.5384615384615384 \][/tex]
So, the probability is [tex]\(0.538 \approx 0.54\)[/tex].
### c) Probability of drawing either a spade or a queen card
Given:
- Total number of cards = 52
- Number of spades = 13
- Number of queens in the deck = 4
- Number of spade queens = 1 (only the queen of spades)
Using the principle of inclusion-exclusion:
[tex]\[ P(\text{Spade or Queen}) = \frac{(\text{Number of Spades} + \text{Number of Queens} - \text{Number of Spade Queens})}{\text{Total Number of Cards}} = \frac{(13 + 4 - 1)}{52} = \frac{16}{52} = 0.3076923076923077 \][/tex]
So, the probability is [tex]\(0.308 \approx 0.31\)[/tex].
### d) Probability of drawing either a heart or a jack
Given:
- Total number of cards = 52
- Number of hearts = 13
- Number of jacks in the deck = 4
- Number of heart jacks = 1 (only the jack of hearts)
Using the principle of inclusion-exclusion:
[tex]\[ P(\text{Heart or Jack}) = \frac{(\text{Number of Hearts} + \text{Number of Jacks} - \text{Number of Heart Jacks})}{\text{Total Number of Cards}} = \frac{(13 + 4 - 1)}{52} = \frac{16}{52} = 0.3076923076923077 \][/tex]
So, the probability is [tex]\(0.308 \approx 0.31\)[/tex].
These are the probabilities for each of the given events:
1. Probability of drawing either a black card or a face card: 0.615
2. Probability of drawing either a red card or an ace: 0.538
3. Probability of drawing either a spade or a queen: 0.308
4. Probability of drawing either a heart or a jack: 0.308
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