Comparison or ordering of two independent values

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1. The probability distribution of a random variable \(X\) is shown in the table, where \(p\) is a constant.

   \(x\)	0	1	2	3
   \(P ( X = x )\)	\(\frac { 1 } { 12 }\)	\(\frac { 1 } { 4 }\)	\(p\)	\(3 p\)

   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 \(\mathrm { P } \left( Y _ { 1 } > Y _ { 2 } \right)\).

9 The probability distribution of a random variable \(X\) is given in the table. Two values of \(X\) are chosen at random. Find the probability that the second value is greater than the first.

The probability distribution of a random variable \(X\) is modelled as follows. $$\text{P}(X = x) = \begin{cases} \frac{k}{x} & x = 1, 2, 3, 4, \\ 0 & \text{otherwise,} \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{12}{25}\). [2]
2. Show in a table the values of \(X\) and their probabilities. [1]
3. The values of three independent observations of \(X\) are denoted by \(X_1\), \(X_2\) and \(X_3\). Find P\((X_1 > X_2 + X_3)\). [3]

In a game, a player notes the values of successive independent observations of \(X\) and keeps a running total. The aim of the game is to reach a total of exactly 7.

1. Determine the probability that a total of exactly 7 is first reached on the 5th observation. [5]