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Sagot :
Certainly! Let's tackle this problem step by step:
### Step-by-Step Solution
1. Establish Variables:
- Let [tex]\( N \)[/tex] be the total number of cards.
- Let [tex]\( G \)[/tex] be the total number of green cards.
2. Probability of Drawing the First Green Card:
- The probability of drawing a green card on the first draw:
[tex]\[ P(\text{First Green}) = \frac{G}{N} \][/tex]
3. Probability of Drawing the Second Green Card Given the First Card is Not Replaced:
- After drawing the first green card, the number of green cards left is [tex]\( G - 1 \)[/tex].
- The total number of cards left is [tex]\( N - 1 \)[/tex].
- The probability of drawing a second green card after drawing the first green card:
[tex]\[ P(\text{Second Green} \mid \text{First Green}) = \frac{G - 1}{N - 1} \][/tex]
4. Combined Probability of Both Events:
- The combined probability of drawing two green cards in succession:
[tex]\[ P(\text{Two Green Cards}) = P(\text{First Green}) \times P(\text{Second Green} \mid \text{First Green}) \][/tex]
[tex]\[ P(\text{Two Green Cards}) = \frac{G}{N} \times \frac{G - 1}{N - 1} \][/tex]
[tex]\[ P(\text{Two Green Cards}) = \frac{G(G - 1)}{N(N - 1)} \][/tex]
5. Simplify the Fraction:
- The fraction [tex]\( \frac{G(G - 1)}{N(N - 1)} \)[/tex] should be simplified to its lowest terms.
- To do this, find the Greatest Common Divisor (GCD) of the numerator and denominator and divide both by that GCD.
### Example with [tex]\( N = 52 \)[/tex] and [tex]\( G = 10 \)[/tex]:
1. Substitute Values:
- Total number of cards, [tex]\( N = 52 \)[/tex]
- Total number of green cards, [tex]\( G = 10 \)[/tex]
2. Calculate the Probability:
[tex]\[ P(\text{Two Green Cards}) = \frac{10 \times 9}{52 \times 51} \][/tex]
[tex]\[ P(\text{Two Green Cards}) = \frac{90}{2652} \][/tex]
3. Simplify the Fraction:
- Find the GCD of 90 and 2652:
[tex]\[ \text{GCD}(90, 2652) = 6 \][/tex]
- Divide both the numerator and the denominator by the GCD:
[tex]\[ \frac{90 \div 6}{2652 \div 6} = \frac{15}{442} \][/tex]
Therefore, the simplified fraction for the probability of drawing two green cards consecutively is [tex]\( \frac{15}{442} \)[/tex].
### Conclusion
- The numerator of the simplified fraction is:
[tex]\[ 15 \][/tex]
Thus, the answer is 15.
### Step-by-Step Solution
1. Establish Variables:
- Let [tex]\( N \)[/tex] be the total number of cards.
- Let [tex]\( G \)[/tex] be the total number of green cards.
2. Probability of Drawing the First Green Card:
- The probability of drawing a green card on the first draw:
[tex]\[ P(\text{First Green}) = \frac{G}{N} \][/tex]
3. Probability of Drawing the Second Green Card Given the First Card is Not Replaced:
- After drawing the first green card, the number of green cards left is [tex]\( G - 1 \)[/tex].
- The total number of cards left is [tex]\( N - 1 \)[/tex].
- The probability of drawing a second green card after drawing the first green card:
[tex]\[ P(\text{Second Green} \mid \text{First Green}) = \frac{G - 1}{N - 1} \][/tex]
4. Combined Probability of Both Events:
- The combined probability of drawing two green cards in succession:
[tex]\[ P(\text{Two Green Cards}) = P(\text{First Green}) \times P(\text{Second Green} \mid \text{First Green}) \][/tex]
[tex]\[ P(\text{Two Green Cards}) = \frac{G}{N} \times \frac{G - 1}{N - 1} \][/tex]
[tex]\[ P(\text{Two Green Cards}) = \frac{G(G - 1)}{N(N - 1)} \][/tex]
5. Simplify the Fraction:
- The fraction [tex]\( \frac{G(G - 1)}{N(N - 1)} \)[/tex] should be simplified to its lowest terms.
- To do this, find the Greatest Common Divisor (GCD) of the numerator and denominator and divide both by that GCD.
### Example with [tex]\( N = 52 \)[/tex] and [tex]\( G = 10 \)[/tex]:
1. Substitute Values:
- Total number of cards, [tex]\( N = 52 \)[/tex]
- Total number of green cards, [tex]\( G = 10 \)[/tex]
2. Calculate the Probability:
[tex]\[ P(\text{Two Green Cards}) = \frac{10 \times 9}{52 \times 51} \][/tex]
[tex]\[ P(\text{Two Green Cards}) = \frac{90}{2652} \][/tex]
3. Simplify the Fraction:
- Find the GCD of 90 and 2652:
[tex]\[ \text{GCD}(90, 2652) = 6 \][/tex]
- Divide both the numerator and the denominator by the GCD:
[tex]\[ \frac{90 \div 6}{2652 \div 6} = \frac{15}{442} \][/tex]
Therefore, the simplified fraction for the probability of drawing two green cards consecutively is [tex]\( \frac{15}{442} \)[/tex].
### Conclusion
- The numerator of the simplified fraction is:
[tex]\[ 15 \][/tex]
Thus, the answer is 15.
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