The probability of rolling a 6 on a biased die is [tex]\(\frac{1}{5}\)[/tex].

a) Complete the tree diagram.

b) Work out the probability of rolling two sixes.



Answer :

Sure, let's solve this step-by-step.

### a) Complete the Tree Diagram

To complete the tree diagram, let's first summarize the probabilities given and derive the necessary probabilities:

- Probability of rolling a 6 on the biased dice, [tex]\( P(6) \)[/tex], is [tex]\( \frac{1}{5} = 0.2 \)[/tex].
- Probability of not rolling a 6 on the biased dice, [tex]\( P(\text{Not } 6) \)[/tex], is:
[tex]\[ 1 - P(6) = 1 - 0.2 = 0.8 \][/tex]

### Tree Diagram for Two Rolls:

1. First Roll:
- Roll a 6: [tex]\( P(6) = 0.2 \)[/tex]
- Not roll a 6: [tex]\( P(\text{Not } 6) = 0.8 \)[/tex]

2. Second Roll, given first roll outcomes:

If the first roll is a 6:
- Roll a 6 again: [tex]\( P(6) = 0.2 \)[/tex]
- Not roll a 6: [tex]\( P(\text{Not } 6) = 0.8 \)[/tex]

If the first roll is not a 6:
- Roll a 6: [tex]\( P(6) = 0.2 \)[/tex]
- Not roll a 6: [tex]\( P(\text{Not } 6) = 0.8 \)[/tex]

The tree diagram branches out as follows:

```
First Roll: Second Roll:
6 (0.2) / \
6 (0.2) Not 6 (0.8)
Not 6 (0.8) / \
6 (0.2) Not 6 (0.8)
```

### b) Work out the probability of rolling two sixes

To find the probability of rolling two sixes in a row, we need to consider the path where both rolls result in a 6. This can be calculated by:

1. First Roll is a 6: [tex]\( P(6) = 0.2 \)[/tex]
2. Second Roll is also a 6: [tex]\( P(6) = 0.2 \)[/tex]

The probability of both events happening successively (rolling a 6 on the first roll and rolling a 6 on the second roll) is calculated by multiplying the probabilities of these independent events:

[tex]\[ P(\text{Two Sixes}) = P(6) \times P(6) = 0.2 \times 0.2 = 0.04 \][/tex]

Therefore, the probability of rolling two sixes in a row is [tex]\( 0.04 \)[/tex] or [tex]\( 4\% \)[/tex].