Suppose that you have 4 green cards and 5 yellow cards. The cards are well shuffled. You randomly draw two cards without replacement. 

G1 = the first card drawn is green
G2 = the second card drawn is green

a.  P(G1 and G2) = 

b.  P(At least 1 green) =

c.  P(G2|G1) = 


d.  Are G1 and G2 independent? 

They are independent events
They are dependent events


To solve the problem, let's denote the total number of cards as \( n = 4 + 5 = 9 \).

### a. \( P(G1 \text{ and } G2) \)

To find the probability that both drawn cards are green:

1. **Probability that the first card is green (G1)**:
   \[
   P(G1) = \frac{4}{9}
   \]

2. **Probability that the second card is green (G2) given that the first was green**:
   After drawing one green card, there are 3 green cards left and 8 cards in total:
   \[
   P(G2 | G1) = \frac{3}{8}
   \]

Now, using the multiplication rule for probabilities:
\[
P(G1 \text{ and } G2) = P(G1) \times P(G2 | G1) = \frac{4}{9} \times \frac{3}{8} = \frac{12}{72} = \frac{1}{6}
\]

### b. \( P(\text{At least 1 green}) \)

To find the probability of drawing at least one green card, it’s easier to calculate the complement, which is the probability of drawing no green cards (i.e., both cards are yellow):

1. **Probability that the first card is yellow**:
   \[
   P(\text{1st yellow}) = \frac{5}{9}
   \]

2. **Probability that the second card is yellow given the first was yellow**:
   After drawing one yellow card, there are 4 yellow cards left and 8 cards total:
   \[
   P(\text{2nd yellow | 1st yellow}) = \frac{4}{8} = \frac{1}{2}
   \]

Now, calculate the probability of both cards being yellow:
\[
P(\text{Both yellow}) = P(\text{1st yellow}) \times P(\text{2nd yellow | 1st yellow}) = \frac{5}{9} \times \frac{1}{2} = \frac{5}{18}
\]

Thus, the probability of at least one green card is:
\[
P(\text{At least 1 green}) = 1 - P(\text{Both yellow}) = 1 - \frac{5}{18} = \frac{13}{18}
\]

### c. \( P(G2 | G1) \)

This was calculated earlier as part of \( P(G1 \text{ and } G2) \):
\[
P(G2 | G1) = \frac{3}{8}
\]

### d. Are \( G1 \) and \( G2 \) independent?

Two events \( A \) and \( B \) are independent if:
\[
P(A \text{ and } B) = P(A) \times P(B)
\]

- We have \( P(G1 \text{ and } G2) = \frac{1}{6} \).
- Calculate \( P(G2) \):
  - The probability that the second card is green can be found by considering all possible cases for the first card:
    1. If the first card is green: \( \frac{4}{9} \times \frac{3}{8} \)
    2. If the first card is yellow: \( \frac{5}{9} \times \frac{4}{8} \)

   Thus,
   \[
   P(G2) = P(G2 | G1) P(G1) + P(G2 | \text{not } G1) P(\text{not } G1) = \left(\frac{3}{8} \cdot \frac{4}{9}\right) + \left(\frac{4}{8} \cdot \frac{5}{9}\right) = \frac{12}{72} + \frac{20}{72} = \frac{32}{72} = \frac{4}{9}
   \]

Now we can check independence:
\[
P(G1) \times P(G2) = \frac{4}{9} \times \frac{4}{9} = \frac{16}{81}
\]

Since \( P(G1 \text{ and } G2) = \frac{1}{6} \) is not equal to \( \frac{16}{81} \), \( G1 \) and \( G2 \) are **dependent events**.

### Summary

- a. \( P(G1 \text{ and } G2) = \frac{1}{6} \)
- b. \( P(\text{At least 1 green}) = \frac{13}{18} \)
- c. \( P(G2 | G1) = \frac{3}{8} \)
- d. **They are dependent events**.

Would you like further details on any part of this problem? Here are some follow-up questions:

1. What would be the probability if there were more cards?
2. How does drawing with replacement change these probabilities?
3. What other combinations of card draws could we analyze?
4. How would the probabilities change if the number of yellow cards increased?
5. Can you explain the concept of independent events further? 

**Tip:** Always consider the total number of outcomes when dealing with probabilities to ensure accuracy.

Suppose that you have 4 green cards and 5 yellow cards. The cards are well shuffled. You randomly draw two cards without replacement. What is P(G1 and G2), P(At least 1 green), P(G2|G1), and are G1 and G2 independent?

Explore the probabilities of drawing cards from a set of green and yellow cards. Learn how to calculate P(G1 and G2), P(At least 1 green), P(G2|G1), and determine the independence of events with detailed explanations and formulas. Suitable for Grades 9-12.

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