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Suppose two cards are drawn randomly. What is the probability of drawing two green cards if the first card is NOT replaced before the second draw? Assume the first card drawn is green.

Show your answer as a fraction in lowest terms.

Enter the numerator:
[?]

Sagot :

GinnyAnswer
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.
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