If the joint probability is the intersection of 2 events, then shouldn't the joint probability of 2 independent events be zero since they don't intersect at all? I'm confused.
There is a difference between
- independent events: $\mathbb P(A \cap B) =\mathbb P(A)\,\mathbb P(B)$, i.e. $\mathbb P(A \mid B)= \mathbb P(A)$ so knowing one happened gives no information about whether the other happened
- mutually disjoint events: $\mathbb P(A \cap B) = 0$, i.e. $\mathbb P(A \mid B)= 0$ so knowing one happened means the other did not happen
You asked for a picture. This might help:
What I understood from your question, is that you might have confused independent events with disjoint events.
disjoint events: Two events are called disjoint or mutually exclusive if they cannot both happen. For instance, if we roll a die, the outcomes 1 and 2 are disjoint since they cannot both occur. On the other hand, the outcomes 1 and “rolling an odd number” are not disjoint since both occur if the outcome of the roll is a 1. The intersect of such events is always 0.
independent events: Two events are independent if knowing the outcome of one provides no useful information about the outcome of the other. For instance, when we roll two dice, the outcome of each is an independent event – knowing the outcome of one roll does not help determining the outcome of the other. Let's build on that example: We roll two dice, a red and a blue. The probability of getting a 1 on the red is given by P(red = 1) = 1/6, and the probability of getting a 1 on the white is given by P(white = 1) = 1/6. It is possible to get their intersect (i.e. both get 1) simply by multiplying them, since they are independent. P(red = 1) x P(white = 1) = 1/6 x 1/6 = 1/36 != 0. In simple words 1/6 of the time the red die is a 1, and 1/6 of those times the white die is 1. To illustrate:
The confusion of the OP lies on the notions of disjoint events and independent events.
One simple and intuitive description of independence is:
A and B are independent if knowing that A happened gives you no information about whether or not B happened.
Or in other words,
A and B are independent if knowing that A happened does not change the probability that B happened.
If A and B are disjoint then knowing that A happened is a game changer! Now you would be certain that B did not happen! And so they are not independent.
The only way independence and "disjointedness" in this example are the same is when B is the empty set (which has probability 0). In this case A happening does not inform anything on B
No pictures but at least some intuition