Finding the probability of whether it would rain on weekends

Question

The probability that it will rain on Saturday is 25% and the probability that it will rain on Sunday is also 25%. Is it true that the probability that it will rain on the weekends is 50%. Explain why or why not.

What i tried

I know that it is not true. Let $P(A)$ represent the probability that it will rain on Saturday, while $P(B)$ represent the probability that it will rain on Sunday. Hence $P(A \cap B)$ will represent the probability that it would rain on both days (weekends) and since $$P(A \cap B)=P(A)+P(B)-P(A\cup B)$$

From the formula above we can see that the sum of $P(A)$ and $P(B)$ alone would not add up to $P(A \cap B)$ which means that we simply could not just add the two probabilities together to get $50$%. Since i find this rather counterintutive could anyone provide a simpler and more clearer explanation to this problem.Thanks

Answer 1

I will copy my comment here and continue:

All the possible outcomes: let former correspond to Saturday and latter correspond to Sunday: $P(rain+rain) = 0.25*0.25$, $P(norain + norain) = 0.75*0.75$, $P(rain+norain) = 0.25*0.75$, $P(norain+rain) = 0.75*0.25$;

The sum = 1 ! Thus probability of rain on both days $= (0.25)^2$, on one of the days $= 2*0.25*0.75 = 0.375$

You are interested in the event: $P(rain+rain)+P(no rain + rain)+P(rain+no rain) = 0.25^2+0.375=0.4375$;

Let's check the formula:

$P(A)=P(B)=0.25$, $P(A\cap B)= P(A)P(B) = 0.25^2$,

$P(A\cup B) = P(A)+P(B)-P(A\cap B) = 0.5 - 0.25^2 = 0.4375$;

Why is this true?

$P(A\cup B) = P(A \backslash B) + P(B \backslash A) + P(A\cap B) = P(rain+norain)+P(norain+rain)+P(rain+rain)$;

Inclusion-exclusion formula that you have used counts $P(A \cap B)$ twice, as

$P(A) = P(A \backslash B) + P(A \cap B)$

$P(B) = P(B \backslash A) + P(A \cap B)$

so it need to be substracted.

Answer 2

You write that "the sum of $P(A)$ and $P(B)$ alone would not add up to $P(A\cup B)$" – but note that the remaining term is subtracted, so it's not just that they don't add up alone, but a probability gets subtracted that's at least $\max(P(A),P(B))$, so that $P(A\cap B)\le\min(P(A),P(B))$.

A more intuitive way to think about this might be that if there's only a $25\%$ chance that it will rain on Saturday, then surely the chance that it will rain on Saturday and on Sunday can't be greater than that – the probability can't increase if you add an extra condition.

Answer 3

You can simply calculate $1$ minus the probability of the complementary event:

$$1-(1-0.25)^2=0.4375$$