Very basic probability question of rules of "and" and "or" I want to start doing Probability from the very basics. However, while going through YouTube videos, I have seen that whenever we have "and", they multiply, and when we have "or", they add. I do not have an intuitive sense as to why is this so.
Can anyone help? Thanks.
 A: Multiplying when you have "AND" is only valid when two events $A,B$ are independent, which is another way of saying the probability of $A$ and $B$ both occurring is $P(A) \cdot P(B)$. Another way of saying it is that knowledge of whether or not $A$ happens has no change on the probability of $B$ occurring, and vice versa. When you have "AND" and the two events are independent, you can ask "Did $A$ happen?" which has probability $P(A)$, and then if $A$ happens you can ask "Did $B$ happen?" which has probability $P(B)$ (due to independence), which is why you multiply. If you want a concrete example, assume you take two fair coins and flip them. Then there are 4 equal probability outcomes, one of which is both heads, and its probability is $1/4$ which is the same as multiplying $(1/2)(1/2)$, i.e. the probabilities of each coin being heads individually.
As for "OR", you can only add if the events are "disjoint," i.e. no way for both events to occur simultaneously. For example, in the two coins example, the event that the first coin is heads and the second coin is tails is disjoint from the event that the first coin is tails and the second coin is heads. Each has probability $1/4$, so the probability that exactly one coin is heads is $1/4 + 1/4 = 1/2$. However the events "first coin is heads" and "second coin is heads" are NOT disjoint, because they can both occur if you get both heads. Thus you can not say that the probability you get at least one head is equal to $1/2 + 1/2 = 1$, which would be what you get if you use addition for OR in this case. 
For a comprehensive treatment of exactly how to handle "OR", including cases where events are not disjoint look up the so-called inclusion-exclusion principle. 
