Independent and mutually exclusive do not mean the same thing.
Two events are independent if one of the following are true:
- \(P(A | B)=P(A)\)
- \(P(B | A)=P(B)\)
- \(P(A \cap B)=P(A) P(B)\)
Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. For example, the outcomes of two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the second roll. To show two events are independent, you must show only one of the above conditions. If two events are NOT independent, then we say that they are dependent.
Sampling may be done with replacement or without replacement.
- With replacement: If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.
- Without replacement: When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.
If it is not known whether A and B are independent or dependent, assume they are dependent until you can show otherwise.
A box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Let \(T\) be the event of getting the white ball twice, \(F\) the event of picking the white ball first, \(S\) the event of picking the white ball in the second drawing.
- Compute \(P(T)\).
- Compute \(P(T|F)\).
- Are \(T\) and \(F\) independent?.
- Are \(F\) and \(S\) mutually exclusive?
- Are \(F\) and \(S\) independent?