Question

3
There are 18 marbles in a bag, all of the same size. The marbles, red, green and blue are in
the ratio 2:3:4 respectively
(a)
€
(b)
Find the number of each colour of the marbles.
If two marbles are selected at random one after the other and replaced,
find the probability that:
(i)
(11)
all are blue;
they are of the same colour.

Answer 1

Alright, let's address each part of the question step-by-step.

### (a) Find the number of each color of marbles.

Given:
- Total marbles = 18
- Ratio of red : green : blue = 2 : 3 : 4

First, we need to find the total ratio:
$\[ \text{Total ratio} = 2 + 3 + 4 = 9 \]$

Now we calculate the number of each color of marbles:

1. Number of red marbles:
$\[ \frac{\text{Ratio of red}}{\text{Total ratio}} \times \text{Total marbles} = \frac{2}{9} \times 18 = 4 \]$

2. Number of green marbles:
$\[ \frac{\text{Ratio of green}}{\text{Total ratio}} \times \text{Total marbles} = \frac{3}{9} \times 18 = 6 \]$

3. Number of blue marbles:
$\[ \frac{\text{Ratio of blue}}{\text{Total ratio}} \times \text{Total marbles} = \frac{4}{9} \times 18 = 8 \]$

So, the number of marbles are:
- Red: 4
- Green: 6
- Blue: 8

### (b) Find the probability that:
#### (i) Both marbles selected are blue

To find the probability of drawing a blue marble, we can use the number of blue marbles:
$\[ \text{Probability of drawing one blue marble} = \frac{\text{Number of blue marbles}}{\text{Total marbles}} = \frac{8}{18} \]$

Since the marbles are replaced after each draw, the probability remains constant. Therefore, the probability of drawing two blue marbles successively (with replacement) is:
$\[ \left(\frac{8}{18}\right)^2 \]$

Calculating this:
$\[ \left(\frac{8}{18}\right)^2 = 0.19753086419753085 \]$

So, the probability that both marbles are blue is approximately 0.1975 or 19.75%.

#### (ii) Both marbles selected are of the same color

To find the probability of drawing two marbles of the same color with replacement, we need to calculate the probabilities for each color and then add them up:

1. Probability of drawing 2 red marbles:
$\[ \left(\frac{4}{18}\right)^2 \]$

2. Probability of drawing 2 green marbles:
$\[ \left(\frac{6}{18}\right)^2 \]$

3. Probability of drawing 2 blue marbles as calculated in part (i):
$\[ \left(\frac{8}{18}\right)^2 \]$

Adding these probabilities together:
$\[ \left(\frac{4}{18}\right)^2 + \left(\frac{6}{18}\right)^2 + \left(\frac{8}{18}\right)^2 = 0.04938271604938271 + 0.1111111111111111 + 0.19753086419753085 = 0.35802469135802467 \]$

Therefore, the probability that both marbles are of the same color is approximately 0.3580 or 35.80%.

### Summary:
(a) Number of each color of marbles:
- Red: 4
- Green: 6
- Blue: 8

(b) Probabilities:
- (i) Both marbles are blue: 0.1975 or 19.75%
- (ii) Both marbles are of the same color: 0.3580 or 35.80%