# Probability of At Least One Event Occurring Question

Suppose on any given day, $$P(A) = 0.37$$ and $$P(B) = 0.21$$, where $$A$$ and $$B$$ are events where:

$$A$$ = Event where the average daily temperature is below $$10$$ degrees

$$B$$ = Event where the daily rainfall is over $$3$$mm

Find the probability of at least one of these events occurring on any given day if there is a $$63\%$$ chance of the daily rainfall being over $$3$$mm, if the temperature is under $$10$$ degrees.

Here's what I've tried:

$$P(B \mid A) = 0.63 \\ P(B \cap A) = P(A) \cdot 0.63 \\ P(B \cap A) = 0.37 \cdot 0.63 = 0.2331$$

However, is this possible as the intersection of $$B$$ and $$A$$ exceeds the probability of $$B$$ itself.

Then I proceed with: \begin{align} P(A \cup B) &= P(A) + P(B) - P(B \cap A) \\ &= 0.37 + 0.21 -0.2331 \\ &=0.3469 \end{align}

Kindly let me know if my calculations and understandings are correct. Thank you!

• You have correctly deduced that the information given in the problem is inconsistent. That implies that any attempted calculation based on the data in the problem will yield only nonsense. Aug 15, 2019 at 5:16