0580

Conditional Probability

Probability

Flashcards

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

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
Worked example

Narrowing the total

Of 40 students, 22 study Spanish. Of those 22, 9 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 22, not 40
  • Of that group, 9 also study German
  • P(German, given Spanish) = 9/22
  • Dividing by 40 would answer a different question — the probability of a student studying both, out of everyone, which is 9/40
  • Writing the restricted total down before the fraction is what makes the difference visible