allo has a bag containing 80 marbles that are either green, yellow or red in colour. $$ \frac{3}{5} $$ of the rbles are green and $$ 10 \% $$ of the marbles are yellow. Paballo picks TWO marbles out of the bag, at a time and without replacing the first one. 1 How many red marbles are in the bag? 2 Draw a tree diagram to represent the above situation. 3 What is the probability that Paballo will choose a GREEN and a YELLOW marble?

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UpStudy AI · Accepted Answer

To solve the problem, let's break it down step by step.

### Step 1: Determine the number of marbles of each color.

1. **Total marbles**: 80
2. **Green marbles**: 
   $$
   \text{Number of green marbles} = \frac{3}{5} \times 80 = 48
   $$
3. **Yellow marbles**: 
   $$
   \text{Number of yellow marbles} = 10\% \times 80 = 0.1 \times 80 = 8
   $$
4. **Red marbles**: 
   $$
   \text{Number of red marbles} = \text{Total marbles} - (\text{Green marbles} + \text{Yellow marbles}) = 80 - (48 + 8) = 80 - 56 = 24
   $$

### Answer to Question 1:
**There are 24 red marbles in the bag.**

### Step 2: Draw a tree diagram.

The tree diagram represents the choices Paballo has when picking two marbles without replacement.

1. **First Pick**:
   - Green (G)
   - Yellow (Y)
   - Red (R)

2. **Second Pick** (depends on the first pick):
   - If the first pick is Green (G):
     - Green (G)
     - Yellow (Y)
     - Red (R)
   - If the first pick is Yellow (Y):
     - Green (G)
     - Yellow (Y)
     - Red (R)
   - If the first pick is Red (R):
     - Green (G)
     - Yellow (Y)
     - Red (R)

The tree diagram can be represented as follows:

```
First Pick
   /   |   \
  G    Y    R
 /|\   /|\   /|\
G Y R G Y R G Y R
```

### Step 3: Calculate the probability of choosing a GREEN and a YELLOW marble.

There are two scenarios for picking a green and a yellow marble:
1. Picking Green first, then Yellow.
2. Picking Yellow first, then Green.

**Scenario 1: Green first, then Yellow**
- Probability of picking Green first:
  $$
  P(G) = \frac{48}{80}
  $$
- After picking a Green marble, there are now 79 marbles left, with 8 of them being Yellow.
- Probability of picking Yellow second:
  $$
  P(Y|G) = \frac{8}{79}
  $$
- Combined probability for this scenario:
  $$
  P(G \text{ then } Y) = P(G) \times P(Y|G) = \frac{48}{80} \times \frac{8}{79} = \frac{48 \times 8}{80 \times 79} = \frac{384}{6320}
  $$

**Scenario 2: Yellow first, then Green**
- Probability of picking Yellow first:
  $$
  P(Y) = \frac{8}{80}
  $$
- After picking a Yellow marble, there are now 79 marbles left, with 48 of them being Green.
- Probability of picking Green second:
  $$
  P(G|Y) = \frac{48}{79}
  $$
- Combined probability for this scenario:
  $$
  P(Y \text{ then } G) = P(Y) \times P(G|Y) = \frac{8}{80} \times \frac{48}{79} = \frac{8 \times 48}{80 \times 79} = \frac{384}{6320}
  $$

### Total Probability:
Now, we add the probabilities of both scenarios:
$$
P(G \text{ and } Y) = P(G \text{ then } Y) + P(Y \text{ then } G) = \frac{384}{6320} + \frac{384}{6320} = \frac{768}{6320}
$$

To simplify:
$$
\frac{768}{6320} = \frac{96}{790} \quad (\text{dividing both by 8})
$$

### Final Answer:
**The probability that Paballo will choose a GREEN and a YELLOW marble is $$ \frac{96}{790} $$.**
There are 24 red marbles in the bag. The probability of choosing a GREEN and a YELLOW marble is $$ \frac{96}{790} $$.

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