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].
