Verify probability from given formula

5 A fair three-sided spinner has sides numbered 1, 2, 3. A fair five-sided spinner has sides numbered \(1,1,2,2,3\). Both spinners are spun once. For each spinner, the number on the side on which it lands is noted. The random variable \(X\) is the larger of the two numbers if they are different, and their common value if they are the same.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 7 } { 15 }\).
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2. Draw up the probability distribution table for \(X\).
3. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

7 Sharma knows that she has 3 tins of carrots, 2 tins of peas and 2 tins of sweetcorn in her cupboard. All the tins are the same shape and size, but the labels have all been removed, so Sharma does not know what each tin contains. Sharma wants carrots for her meal, and she starts opening the tins one at a time, chosen randomly, until she opens a tin of carrots. The random variable \(X\) is the number of tins that she needs to open.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 6 } { 35 }\).
2. Draw up the probability distribution table for \(X\).
3. Find \(\operatorname { Var } ( X )\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 Anil is taking part in a tournament. In each game in this tournament, players are awarded 2 points for a win, 1 point for a draw and 0 points for a loss. For each of Anil's games, the probabilities that he will win, draw or lose are \(0.5,0.3\) and 0.2 respectively. The results of the games are all independent of each other. The random variable \(X\) is the total number of points that Anil scores in his first 3 games in the tournament.

1. Show that \(\mathrm { P } ( X = 2 ) = 0.114\).
2. Complete the probability distribution table for \(X\).

   \(x\)	0	1	2	3	4	5	6
   \(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)			0.114	0.207	0.285		0.125

3. Find the value of \(\operatorname { Var } ( X )\).

3 Jeremy is a computing consultant who sometimes works at home. The number, \(X\), of days that Jeremy works at home in any given week is modelled by the probability distribution $$\mathrm { P } ( X = r ) = \frac { 1 } { 40 } r ( r + 1 ) \quad \text { for } r = 1,2,3,4 .$$

1. Verify that \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 2 }\).
2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Jeremy works for 45 weeks each year. Find the expected number of weeks during which he works at home for exactly 2 days.

15 In this question you must show detailed reasoning. The random variable \(X\) has probability distribution defined as follows.
\(\mathrm { P } ( X = x ) = \begin{cases} \frac { 15 } { 64 } \times \frac { 2 ^ { x } } { x ! } & x = 2,3,4,5 ,
0 & \text { otherwise. } \end{cases}\)

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 15 } { 32 }\). The values of three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\).
2. Given that \(X _ { 1 } + X _ { 2 } + X _ { 3 } = 9\), determine the probability that at least one of these three values is equal to 2 . Freda chooses values of \(X\) at random until she has obtained \(X = 2\) exactly three times. She then stops.
3. Determine the probability that she chooses exactly 10 values of \(X\). \section*{END OF QUESTION PAPER} \section*{OCR} Oxford Cambridge and RSA