Question

In independent events, an event does not get affected by another, and as such, the probability of an event, when another event occurs, does not change. For example, when calculating coin tosses, event B does not alter event A, as these are independent. The formula for conditional probability in dependent events is P(A|B) = P(A ∩ B)/P(B).
Example of dependent events:
Drawing two cards without replacement in a regular deck. For example, the probability of an event B, when event A has already happened, is 3/51, while the probability of an event B, in general, is 4/52.

Answer 1

You’re on the right track. Key points and formulas:

- Conditional probability:
P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.

- Independence:
A and B are independent iff knowing that B occurred does not change the probability of A:
P(A|B) = P(A). Equivalently,
P(A ∩ B) = P(A)·P(B).

Card example (A = “first card is an Ace”, B = “second card is an Ace”) without replacement:
- P(B) (unconditional probability that a random card is an Ace) = 4/52 = 1/13.
- P(B|A) (probability second is an Ace given first was an Ace) = 3/51 = 1/17.
These are not equal, so the two events are dependent. Also
P(A ∩ B) = (4/52)·(3/51) = 12/2652 = 1/221.

If you drew with replacement (put the first card back and reshuffle), then P(B|A) = 4/52 = 1/13 = P(B), so the draws would be independent.