1 In a quiz a contestant is asked up to four questions. The contestant's turn ends once the contestant gets a question wrong or has answered all four questions. The probability that a particular contestant gets any question correct is 0.6 , independently of other questions. The discrete random variable \(X\) models the number of questions which the contestant gets correct in a turn.

1. Show that \(\mathrm { P } ( X = 4 ) = 0.1296\). The probability distribution of \(X\) is shown in Fig. 1.1. \begin{table}[h]

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	0.4	0.24	0.144	0.0864	0.1296

   \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{table}
2. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   The number of points that a contestant scores is as shown in Fig. 1.2. \begin{table}[h]

   Number of questions correct	Number of points scored
   0 or 1	0
   2	2
   3	3
   4	5

   \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{table} The discrete random variable \(Y\) models the number of points which the contestant scores.
3. Without doing any working, explain whether each of the following will be less than, equal to or greater than the corresponding value for \(X\).
   • \(\mathrm { E } ( Y )\)
   • \(\operatorname { Var } ( Y )\)