I came across the following question in a textbook (bear in mind that this is the only information given)-
There is a $50$ percent chance of rain today. There is a $60$ percent chance of rain tomorrow. There is a $30$ percent chance that it will not rain either day. What is the chance that it will rain both today and tomorrow?
My instinct was to multiply the probabilities together for today and tomorrow to arrive at an answer of $30$ percent. However, the answer given is $40$ percent (based on taking the addition of the individual probabilities and subtracting their union). Can someone explain to me why the multiplication rule does not apply here? Does it have to do with independence? Bear in mind, I am trying to relearn probability theory from scratch. Thanks.
Best Answer
Yes, this is about independence. Or, better put, these events are not independent, which by common sense makes sense, since if it rains one day, it typically means there is a higher chance than normal that it rains the next day as well (we might well be in the 'rainy season')
You can also tell that the events are not independent given the numbers given to you. If the events were independent, then the probability of it not raining either day should have been $0.5 \cdot 0.4=0.2$, but you were told it is actually $0.3$.
What is always true, however (so you can always use this formua, whether the events are independent or not), is that:
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
And from that you can derive your desired:
$$P(A \cap B) = P(A) +P(B) - P(A \cup B) = P(A) +P(B)-(1-P(A^C \cap B^C))=$$
$$0.5+0.6-(1-0.3) =1.1-0.7=0.4$$