1. Conditional Probability and Independent Events
Section 7.5
Conditional Probability and Independent Events

2. Conditional Probability
When the probability of an event is affected by the knowledge of other information relevant to the event.
Notation: P(A|B) is read “the probability of event A given that event B has occurred.”
Ex. You roll a fair die. Find the probability that you roll a 2 given that your roll is an even.
Knowing it is even restricts the sample space to {2, 4, 6}. So

3. Conditional Probability of an Event
If A and B are events in an experiment and then the conditional probability that the event B will occur given that A has already occurred is
Which can be written:

4. Ex. In a box of 20 size AA batteries, 10 are brand X and 10 are brand Y. You also know that 3 of the brand X batteries are dead, while 2 of the brand Y are dead. Find the probability that in a (random) draw
a. you get a dead (D) brand X battery.
b. you get brand Y given that you drew a dead (D) battery.

5. Test for Independent Events
If A and B are independent events, then
Test for Independent Events
Events A and B are independent events if and only if
Note: this generalizes to more than two independent events.

6. Therefore V and R are independent
Ex. If A die is rolled twice, show that rolling a 5 on the first roll and rolling a 4 on the second roll are independent events.
(roll 1, roll2)
V = 5 on first roll, R = 4 on second roll
Therefore V and R are independent