1.2.2 Independence

Question 1

What does it mean for two events A and B to be independent?

Answer 1

The occurrence of event A does not affect the probability of event B, and vice versa. Mathematically, P(A ∩ B) = P(A)P(B).

Question 2

What equation characterizes independence using conditional probability?

Answer 2

If A and B are independent, then P(A | B) = P(A) and P(B | A) = P(B).

Question 3

Is the converse of the independence definition true?

Answer 3

Yes. If P(A | B) = P(A), then A and B are independent.

Question 4

How is mutual exclusivity different from independence?

Answer 4

Mutually exclusive events cannot both occur (P(A ∩ B) = 0), while independent events can both occur, but the occurrence of one does not affect the other.

Question 5

Can events be both disjoint and independent?

Answer 5

Only if at least one event has zero probability. Otherwise, disjoint (mutually exclusive) events are always dependent.

Question 6

What is the probability of two independent events A and B both occurring?

Answer 6

P(A ∩ B) = P(A)P(B)

Question 7

In the context of rolling two dice, are the outcomes of each roll independent?

Answer 7

Yes. The result of the first die roll does not affect the result of the second.

Question 8

How is independence used in multistep probability problems like drawing from two boxes?

Answer 8

If the outcomes from different steps are independent, joint probabilities are calculated by multiplying individual probabilities.