Definitions:
- If events A and B are mutually exclusive, both cannot occur at the same time. For example, if event A occurs, event B will not occur too, and vice versa. On the other hand, if both events are not mutually exclusive, P(A \cap B) > 0 or in other words, there is some non-zero chance that events A and B can occur at the same time.
- If events A and B are independent, the occurrence of one does not affect the probability of the other. For example, getting heads in a fair coin in the first flip does not affect the probability of getting tails in the next flip, and vice versa. Furthermore, when independent, P(A \cap B) = P(A) \times P(B). On the other hand, if both events are not independent, the occurrence of event A will change the probability of event B, so that P(B) \neq 0.5.
A quick search on Google and Claude showed that the answer is no. Here’s me paraphrasing an explanation from storyofmathematics.com: if events A and B are mutually exclusive, it means that the occurrence of A will cause the probability of B to be zero and vice versa. Hence, they cannot be independent.
Here’s another explanation from stackexchange.com: there’s a special case where events A and B can be both mutually exclusive and independent, which is when P(A) \times P(B) = 0, meaning P(A) or P(B) or both are zero.
But none of the above explanations cleared my confusion. Here’s one example I thought of that’s throwing me off.
- When throwing ONE fair die TWICE, the events of getting 1 first does not affect the probability of getting 2 in the second throw. So they’re independent.
- When throwing ONE fair die, it’s impossible to get 1 and 2 at the same time. So they’re mutually exclusive.
- However, when throwing ONE fair die TWICE, the events of getting 1 and getting 2 are now NOT mutually exclusive. It’s possible to get 1 and 2, albeit NOT exactly at the same time, but sort of in “one chunk of time”.
But from what I’ve learned, it seems like throwing two dices once is different than throwing one dice twice. How is that so?