Conditional Probability
What is conditional probability?
- Conditional probability is where the probability of an event happening can vary depending on the outcome of a prior event
- The event A happening given that event B has happened is denoted A|B
- A common example of conditional probability involves selecting multiple objects from a bag without replacement
- The probability of selecting a certain item changes depending on what was selected before
- This is because the total number of items will change as they are not replaced once they have been selected
How do I calculate conditional probabilities?
- Some conditional probabilities can be calculated by using counting outcomes
- Probabilities without replacement can be calculated like this
- For example: There are 10 balls in a bag, 6 of them are red, two of them are selected without replacement
- To find the probability that the second ball selected is red given that the first one is red count how many balls are left:
- A red one has already been selected so there are 9 balls left and 5 are red so the probability is
- You can use sample space diagrams to find the probability of A given B:
- reduce your sample space to just include outcomes for event B
- find the proportion that also contains outcomes for event A
- There is a formula for conditional probability that you should use
- This is given in the formula booklet
- This can be rearranged to give
- By symmetry you can also write
How do I tell if two events are independent using conditional probabilities?
- If and are two events then they are independent if:
- Equally you can still use to test for independence
- This is given in the formula booklet
Worked Example
Let be the event that it is raining in Weatherville and be the event that there is a thunderstorm in Weatherville.
It is known that , and .
a)
Find the probability that it is raining in Weatherville.
b)
State whether the events and are independent. Give a reason for your answer.