Box ,4 contains 6 red cards, 4 blue cards and 2 green cards.
Box.R contains 3 red cards and 5 blue cards.
A card is drarrn at random from Box I and put into Box B.
Next, a card is drawn at random from Box B.
Find, as a fraction in its simplest fom, the probability that
(i) two green cards are drawn,

Let's define the problem clearly and step-by-step. There are two boxes: 

- **Box I**: Contains 6 red cards, 4 blue cards, and 2 green cards.
- **Box B**: Contains 3 red cards and 5 blue cards.

### Part (i): Probability of drawing two green cards

First, we observe that **Box B** has no green cards initially, so the only way to draw a green card from Box B is if a green card is transferred from Box I to Box B in the first step.

### Step 1: Drawing a green card from Box I
Box I has a total of:
\[
6 \text{ (red)} + 4 \text{ (blue)} + 2 \text{ (green)} = 12 \text{ cards}.
\]
The probability of drawing a green card from Box I is:
\[
P(\text{green from I}) = \frac{2}{12} = \frac{1}{6}.
\]

### Step 2: Drawing a green card from Box B after transfer
If a green card is transferred from Box I to Box B, Box B will now contain 1 green card, 3 red cards, and 5 blue cards, for a total of 9 cards. The probability of drawing this green card from Box B is:
\[
P(\text{green from B}) = \frac{1}{9}.
\]

### Combined Probability
The combined probability of drawing a green card from Box I and then a green card from Box B is the product of these two probabilities:
\[
P(\text{two green cards}) = P(\text{green from I}) \times P(\text{green from B}) = \frac{1}{6} \times \frac{1}{9} = \frac{1}{54}.
\]

Thus, the probability that two green cards are drawn is:
\[
\boxed{\frac{1}{54}}.
\]

Would you like more details or have any questions about this? Here are 5 related questions to expand on this concept:

1. What is the probability of drawing two red cards?
2. How would the probability change if Box B initially had green cards?
3. What is the probability of drawing a red card first, then a blue card?
4. Can we calculate the total probability of drawing any green card from Box B?
5. How does adding more green cards to Box I affect the probability?

**Tip**: When calculating sequential probabilities, multiply the probabilities of each independent event.

Box I contains 6 red cards, 4 blue cards and 2 green cards. Box B contains 3 red cards and 5 blue cards. A card is drawn at random from Box I and put into Box B. Next, a card is drawn at random from Box B. Find, as a fraction in its simplest form, the probability that two green cards are drawn.

Find the probability of drawing two green cards from Box I and Box B in a step-by-step manner. This problem involves probability concepts and the multiplication rule. Suitable for students in Grades 9-11.

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