Comparison or ordering of two independent values

Questions asking about the relationship between two independent values, such as which is larger, whether they differ, or conditional probabilities based on their relative ordering.

3 questions

OCR PURE Q13
13
  1. The probability distribution of a random variable \(X\) is shown in the table, where \(p\) is a constant.
    \(x\)0123
    \(P ( X = x )\)\(\frac { 1 } { 12 }\)\(\frac { 1 } { 4 }\)\(p\)\(3 p\)
    Two values of \(X\) are chosen at random. Determine the probability that their product is greater than their sum.
  2. A random variable \(Y\) takes \(n\) values, each of which is equally likely. Two values, \(Y _ { 1 }\) and \(Y _ { 2 }\), of \(Y\) are chosen at random. It is given that \(\mathrm { P } \left( Y _ { 1 } = Y _ { 2 } \right) = 0.02\).
    Find \(\mathrm { P } \left( Y _ { 1 } > Y _ { 2 } \right)\).
OCR MEI AS Paper 2 2018 June Q4
4 The probability distribution of the discrete random variable \(X\) is given in Fig. 4 . \begin{table}[h]
\(r\)01234
\(\mathrm { P } ( X = r )\)0.20.150.3\(k\)0.25
\captionsetup{labelformat=empty} \caption{Fig. 4}
\end{table}
  1. Find the value of \(k\).
    \(X _ { 1 }\) and \(X _ { 2 }\) are two independent values of \(X\).
  2. Find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 6 \right)\).
OCR AS Pure 2017 Specimen Q9
9 The probability distribution of a random variable \(X\) is given in the table.
\(x\)123
\(\mathrm { P } ( X = x )\)0.60.30.1
Two values of \(X\) are chosen at random. Find the probability that the second value is greater than the first.