Let's address each part of the question step-by-step.
### Part (a)
Work out the probability that it will rain on all three days.
Given:
- Probability of rain on a single day (Friday, Saturday, or Sunday) = 70% = 0.7
To find the probability that it rains on all three days, we multiply the probabilities for each day:
[tex]\[
P(\text{Rain on all three days}) = P(\text{Rain on Friday}) \times P(\text{Rain on Saturday}) \times P(\text{Rain on Sunday})
\][/tex]
[tex]\[
P(\text{Rain on all three days}) = 0.7 \times 0.7 \times 0.7 = 0.343
\][/tex]
Thus, the probability that it will rain on all three days is approximately 0.343.
### Part (b)
Work out the probability that it will rain on exactly two consecutive days.
To solve this, first we need to consider various scenarios and then sum up their probabilities. There are exactly three possible scenarios where it rains on exactly two days out of three:
1. It rains on Friday and Saturday, but not on Sunday.
2. It rains on Friday and Sunday, but not on Saturday.
3. It rains on Saturday and Sunday, but not on Friday.
Given:
- Probability of rain on a single day (Friday, Saturday, or Sunday) = 0.7
- Probability of no rain on a single day = 1 - 0.7 = 0.3
For each scenario involving exactly two consecutive rainy days:
[tex]\[
P(\text{Rain on exactly two consecutive days}) = P(\text{Two rainy days}) \times P(\text{One no-rainy day})
\][/tex]
Now calculate for each of the three scenarios:
1. Rain on Friday and Saturday, but not on Sunday:
[tex]\[
= 0.7 \times 0.7 \times 0.3
\][/tex]
2. Rain on Friday and Sunday, but not on Saturday:
[tex]\[
= 0.7 \times 0.3 \times 0.7
\][/tex]
3. Rain on Saturday and Sunday, but not on Friday:
[tex]\[
= 0.3 \times 0.7 \times 0.7
\][/tex]
Since each probability calculation results in:
[tex]\[
0.147
\][/tex]
There are three such scenarios, so we sum them:
[tex]\[
P(\text{Exactly two consecutive days}) = 3 \times 0.147 = 0.441
\][/tex]
Thus, the probability that it will rain on exactly two out of three days is approximately 0.441.
