What are two ways to determine the probability that two independent events, A and B, will occur together.

Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring.

Some other examples of independent events are:

Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die. Choosing a marble from a jar AND landing on heads after tossing a coin.

The events A and B are independent if any one of the following three equivalent conditions hold:

P (A ∩ B) = P (A) P (B)

P (A|B) = P (A) B has no effect on A

P (B|A) = P (B) A has no effect on B

Intuitively, two events are independent if the occurrence of one has no effect at all on the probability of the other.

To find the probability of two independent events that occur in sequence, find the probability of each event occurring separately, and then multiply the probabilities.

Multiplication Rule: When two events, A and B, are independent, the probability of both occurring is:

P (A and B) = P (A) · P (B)

To test whether two events A and B are independent, calculate P (A), P (B), and P (A ∩ B), and then check whether P (A ∩ B) equals P (A) P (B). If they are equal, A and B are independent; if not, they are dependent.