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

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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 5 over 9

  • 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

    • space straight P left parenthesis A vertical line B right parenthesis equals fraction numerator straight P stretchy left parenthesis A intersection B stretchy right parenthesis over denominator straight P stretchy left parenthesis B stretchy right parenthesis end fraction

    • 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 Aand B are two events then they are independent if:

    • straight P left parenthesis A vertical line B right parenthesis equals straight P left parenthesis A right parenthesis equals straight P left parenthesis A vertical line B apostrophe right parenthesis

  • Equally you can still use straight P left parenthesis A intersection B right parenthesis equals straight P left parenthesis A right parenthesis straight P left parenthesis B right parenthesis to test for independence

    • This is given in the formula booklet

Worked Example

Let R be the event that it is raining in Weatherville and T be the event that there is a thunderstorm in Weatherville.

It is known that straight P left parenthesis T right parenthesis equals 0.035, straight P left parenthesis T intersection R right parenthesis equals 0.03 and straight P left parenthesis T vertical line R right parenthesis equals 0.15.

a) Find the probability that it is raining in Weatherville.

4-3-2-ib-aa-sl-conditional-prob-a-we-solution

b) State whether the events R and T are independent. Give a reason for your answer.

4-3-2-ib-aa-sl-conditional-prob-b-we-solution
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Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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