Independence, Mutual Exclusivity, and Conditional Probability

All flashcards

What are independent events?

Events are independent when knowing that one occurred does not change the probability that the other occurs.

What equation tests independence using a joint probability?

For independent events $A$ and $B$, $P(A \text{ AND } B)=P(A)P(B)$.

How can conditional probabilities show independence?

Check whether $P(A|B)=P(A)$ or $P(B|A)=P(B)$. Either equality is sufficient to establish independence.

Why are draws with replacement treated as independent?

Returning the selected item restores the original population, so one draw does not change the probabilities for the next draw.

Why are draws without replacement dependent?

A selected item is removed, changing the population and the probabilities for later selections.

Quiz questions

1. What characterizes mutually exclusive events?Answer: They have no outcomes in common.
2. Which condition means that events $A$ and $B$ are mutually exclusive?Answer: $P(A \text{ AND } B)=0$
3. If mutually exclusive events $A$ and $B$ both have positive probability, how are they related?Answer: They are dependent because either event rules out the other.
4. For mutually exclusive events $A$ and $B$, which formula gives the probability that at least one occurs?Answer: $P(A \text{ OR } B)=P(A)+P(B)$
5. Which equation is a valid test for independence of $A$ and $B$?Answer: $P(A \text{ AND } B)=P(A)P(B)$