MathematicsExam code: 0580
Conditional Probability
Probability
A restricted total
- When one event is known to have occurred already, the chance of a second event is a conditional probability, signalled by the words given that
- Knowing that the first thing happened rules out some of the outcomes, so the calculation runs over a smaller set than usual
- Everything else stays the same: it is still successes divided by total, but the total is now only the outcomes that meet the condition
- The phrase to watch for is "given that", and it is almost always how these questions are introduced
- A useful way to think about it: the condition tells you which part of the diagram to look inside, and you never leave that part
Worked example
Choosing from a smaller group
Of 36 students, 19 study Spanish. Of those 19, 7 also study German. A student is chosen at random from those who study Spanish. Find the probability that the student also studies German.
Solution:
- The words "from those who study Spanish" restrict the group being chosen from
- The total is therefore 19, not 36
- Of that group, 7 also study German
- P(German, given Spanish) = 7/19
- Dividing by 36 would answer a different question: the probability of a student studying both, out of everyone, which is 7/36
- Writing the restricted total down before the fraction is what makes the difference visible
What is and is not required
- Neither the notation P(A|B) nor any conditional-probability formula is required by this syllabus, which says so in as many words
- What is required is calculating them from a Venn diagram, a tree diagram or a table
- So the whole topic is about reading the right region of a diagram rather than applying anything memorised
- Recognising the notation does no harm, but nothing is lost by not knowing it