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Statistics & Probability

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Topic overview

Discrete/continuous distributions, conditional probability, expectation and variance

Key concepts

Verify STEP 2/3 separately. Complex numbers and 2×2 matrices may belong to STEP 2; De Moivre, general complex roots, planes and vector products, etc. belong to STEP 3.

Key points

  • ◆Follow the prerequisite knowledge requirements given by the examination body; new concepts may only be used when the question provides sufficient guidance.
  • ◆The examples on this site are training with a limited scope and do not represent full syllabus coverage or official difficulty.

Worked examples

Try each example yourself before checking the explanation.

1Written examples within the scope

A bag contains rr red balls and bb blue balls, where r,b≥1r,b \ge 1. Two balls are drawn at random without replacement.

Find the probability that both balls are the same colour.

Show that the probability that both are the same colour equals r2+b2−r−b(r+b)(r+b−1)\dfrac{r^2+b^2-r-b}{(r+b)(r+b-1)}.

Find, in terms of rr and bb, the expected number of red balls drawn.

Explain intuitively why E[X]=2rr+b\mathbb{E}[X] = \dfrac{2r}{r+b} makes sense.

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