Questions — SPS SPS SM Statistics (77 questions)

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SPS SPS SM Statistics 2026 January Q8
8. The discrete random variable \(R\) takes even integer values from 2 to \(2 n\) inclusive.
The probability distribution of \(R\) is given by $$\mathrm { P } ( R = r ) = \frac { r } { k } \quad r = 2,4,6 , \ldots , 2 n$$ where \(k\) is a constant.
  1. Show that \(k = n ( n + 1 )\) When \(n = 20\)
  2. find the exact value of \(\mathrm { P } ( 16 \leqslant R < 26 )\) When \(n = 20\), a random value \(g\) of \(R\) is taken and the quadratic equation in \(x\) $$x ^ { 2 } + g x + 3 g = 5$$ is formed.
  3. Find the exact probability that the equation has no real roots.
SPS SPS SM Statistics 2026 January Q9
9. The Venn diagram, where \(p , q\) and \(r\) are probabilities, shows the events \(A , B , C\) and \(D\) and associated probabilities.
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  1. State any pair of mutually exclusive events from \(A\), \(B\), \(C\) and \(D\) The events \(B\) and \(C\) are independent.
  2. Find the value of \(p\)
  3. Find the greatest possible value of \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\) Given that \(\mathrm { P } \left( B \mid A ^ { \prime } \right) = 0.5\)
  4. find the value of \(q\) and the value of \(r\)
  5. Find \(\mathrm { P } \left( [ A \cup B ] ^ { \prime } \cap C \right)\)
  6. Use set notation to write an expression for the event with probability \(p\)