Consider 3 events and their probabilities:
Probability that today is Friday: $P(F)= \frac{1}{7}$
Probability that today is Saturday: $P(S) = \frac{1}{7}$
Probability that it is the weekend: $P(W) = \frac{2}{7}$
I know the probability that it is Friday or the weekend is just $P(F)$+$P(W)$. But, if I wanted to find the probability that it is Saturday or the weekend which are overlapping events, can $P(S \cup W)=P(S)+P(W)-P(S \cap W)$ be used? Because when I think about it, $P(S \cap W)$ should be $\frac{1}{7}$ but $\frac{1}{7}$ is not equal to $P(S) * P(W) = \frac{2}{49}$. Is there a fundamental rule I'm overlooking?
Best Answer
$P(S \cap W)$ is only equal to $P(S) * P(W)$ if the events $S$ and $W$ are independent, but in this case $S$ and $W$ are not independent.