Independent and Conditional Events

The conditional probability of given is

which is the probability of given has already occurred.

A bag contains 3 white balls, marked with 1, 2, or 3, a red ball marked with 4, and 4 blue balls marked with 5, 6, 7, and 8.

1. What is the probability of selecting a white ball

2. What is the probability selecting a white ball given that the ball selected had an odd number

The probability that Marianne will go to Western is . The probability that Marianne's boyfriend will go to Western is . If Marianne goes to Western, the probability that her boyfriend will go to Western is What is the probability they both go to Western

If and are independent events (that is, they are mutually exclusive events), then

If and are independent events, then:

1. and are independent
2. and are independent
3. and are independent

Consider two fair coin tosses , , and defined as:
A: first toss is a head
B: second toss is a head
C: both tosses are the same

A and B: independent
A and C: independent
B and C: independent

So A, B, C, are dependent.

Since and since for independent events:

Two events and are independent if and only if:

Similarly:
and

Two cards are drawn from a well-shuffled deck of cards. A heart face card is drawn, and then a diamond is drawn. Calculate the probabilities if:

1. The first card is replaced

2. The first card is not replaced