Conditional Probability (DP IB Analysis & Approaches (AA)): Revision Note

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