Conditional Probability (DP IB Maths: AI HL)

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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 can 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 

Worked example

In a class of 30 students: 19 students have a dog, 17 students have a cat and 11 have both a dog and a cat. One student is selected at random.

a)
Find the probability that the student has a dog.

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

b)
Find the probability that the student has a dog given that they have a cat.

4-3-2-ib-ai-sl-conditional-prob-b-we-solution

c)
Find the probability that the student has a cat given that they have a dog.

4-3-2-ib-ai-sl-conditional-prob-c-we-solution

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Dan

Author: Dan

Expertise: Maths

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.