Answer:
Part one: 1/8
Part two: 7/8
Explanation:
We can use the notatios (s,s,s) to denote that all three days are sunny, (s,r,s) to denote that the first day is sunny, the second is rainy and the third one is sunny, etc.
Using the notation above, we can describe the outcome of out trip with a sample space, that is, a set with all the possible results our trip will have.
The sample space is Ω = {(s,s,s) ; (r,s,s) ; (s,r,s) ; (s,s,r) ; (s,r,r) ; (r,s,r) ; (r,r,s) ; (r,r,r) }. Note that Ω has a total of 8 possible events that can occur and all of them with equal probability, because the probability of raining is the same than the probability of not raining. We can calculate the probability of certain event by dividing the amount favourable cases that match what the event describes with the total amount of cases from Ω, that is, 8.
PART ONE: We have only one favourable element (s,s,s). Since the probability from each element of Ω is the same, we conclude that the probability that all three days is sunny is 1/8
PART TWO: Any element of Ω that contains a rainy day (an 'r') is a favourable element. Thus, any element different from (s,s,s) is a favourable element, giving us 7 favourable cases from a total of 8. Thus, the probability is 7/8.
We could have calculated this by realizing that having al laest one rainy day is the contrary from having all three days sunny. In order to calculate the probability we can calculate instead the probability of the complementary event and substract it from 1. This is useful because we know that the probability that it is sunny all three days is 1/8, thus the probability of having at least one rainy day is 1-1/8 = 7/8.
I hope this helped you!