Question

Spring in Auckland tends to have very changeable weather.

In one particular spring suppose that at least some rain was recorded on 96% of days, no wind was recorded on 10% of days and only 3% of days recorded no wind or rain.
Let Ω = {daily spring weather in Auckland} and define two events R and W on this sample space.
Let R be the event that rain was recorded in Auckland on a randomly selected spring day,
and W be the event that wind was recorded in Auckland on a randomly selected spring day.

A) Using the information given we could write the probability statement P(R) = 0.96. Write
down the rest of the information given in terms of two more probability statements

B) Find the probability that wind was recorded on a randomly chosen spring day.

C) Find the probability that at least one of wind or rain was recorded on a randomly chosen
spring day.

D) Find the probability that both wind and rain were recorded on a randomly chosen spring
day

E) Find the probability that rain but no wind was recorded on a randomly chosen spring day.

F) If wind was recorded on a randomly chosen spring day find the probability that rain was
recorded on the same day

Answer 1

A detailed explanation of probability statements related to rain and wind in Auckland during spring, along with calculations for various probabilities.

Step-by-step explanation:

A) Using the information given, we can write the probability statement P(R) = 0.96.

We are also given that the probability of no wind being recorded on a spring day is 0.10 (P(W') = 0.10) and the probability of no wind or rain being recorded is 0.03 (P(R' ∩ W') = 0.03).

B) To find the probability that wind was recorded on a randomly chosen spring day, we can use the complement rule. P(W) = 1 - P(W') = 1 - 0.10 = 0.90.

C) To find the probability that at least one of wind or rain was recorded on a randomly chosen spring day, we can use the principle of inclusion-exclusion. P(R ∪ W) = P(R) + P(W) - P(R ∩ W) = 0.96 + 0.90 - 0.03 = 1.83 - 0.03 = 0.83.

D) To find the probability that both wind and rain were recorded on a randomly chosen spring day, we can use the intersection rule. P(R ∩ W) = P(R) + P(W) - P(R ∪ W) = 0.96 + 0.90 - 0.83 = 0.06.

E) To find the probability that rain but no wind was recorded on a randomly chosen spring day, we can use the difference rule. P(R ∩ W') = P(R) - P(R ∩ W) = 0.96 - 0.06 = 0.90.

F) If wind was recorded on a randomly chosen spring day, the probability that rain was recorded on the same day can be found using the conditional probability formula. P(R|W) = P(R ∩ W) / P(W) = 0.06 / 0.90 = 0.067.