Three or more independent values

3 A factory makes a large number of ropes with lengths either 3 m or 5 m . There are four times as many ropes of length 3 m as there are ropes of length 5 m .

1. One rope is chosen at random. Find the expectation and variance of its length.
2. Two ropes are chosen at random. Find the probability that they have different lengths.
3. Three ropes are chosen at random. Find the probability that their total length is 11 m .

1 The table shows the probability distribution of a random variable \(X\).

1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
2. Three values of \(X\) are chosen at random. Find the probability that \(X\) takes the value 2 at least twice.

11 Two girls, Lili and Hui, play a game with a fair six-sided dice. The dice is thrown 10 times. \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 10 }\) represent the scores on the \(1 ^ { \text {st } } , 2 ^ { \text {nd } } , \ldots , 10 ^ { \text {th } }\) throws of the dice. \(L\) denotes Lili's score and \(L = 10 X _ { 1 }\). \(H\) denotes Hui's score and \(H = X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 }\).

1. Calculate
   The spreadsheet below shows a simulation of 25 plays of the game. The cell E3, highlighted, shows the score when the dice is thrown the fourth time in the first game. \begin{table}[h]

   	A	B	C	D	E	F	G	H	I	J	K	L	M	N
   1	Throw of dice												Lili's	Hui's
   2		1	2	3	4	5	6	7	8	9	10		score	score
   3	Game 1	3	5	2	1	1	3	1	1	1	4		30	22
   4	Game 2	6	3	2	4	4	3	5	3	3	5		60	38
   5	Game 3	6	4	2	6	5	2	1	5	2	3		60	36
   6	Game 4	1	5	1	6	6	3	1	4	6	2		10	35
   7	Game 5	4	4	3	1	6	4	4	1	6	2		40	35
   8	Game 6	2	1	5	1	2	5	1	5	2	3		20	27
   9	Game 7	1	1	3	4	4	5	6	3	4	2		10	33
   10	Game 8	1	1	3	6	3	4	4	5	2	3		10	32
   11	Game 9	2	2	2	4	3	2	1	5	5	6		20	32
   12	Game 10	3	5	3	3	5	3	4	3	1	1		30	31
   13	Game 11	5	3	6	5	5	4	2	1	1	5		50	37
   14	Game 12	6	4	3	2	4	1	3	3	5	3		60	34
   15	Game 13	2	3	2	1	2	2	2	2	2	1		20	19
   16	Game 14	4	1	3	3	1	2	6	6	1	3		40	30
   17	Game 15	5	1	2	6	3	4	6	3	6	4		50	40
   18	Game 16	3	6	1	1	5	3	1	3	3	3		30	29
   19	Game 17	5	2	5	2	4	5	2	2	3	4		50	34
   20	Game 18	3	6	3	5	5	2	3	1	1	2		30	31
   21	Game 19	6	6	3	1	5	6	3	4	1	6		60	41
   22	Game 20	2	6	4	5	6	5	2	4	3	3		20	40
   23	Game 21	5	3	5	4	5	3	3	6	6	1		50	41
   24	Game 22	6	3	5	5	6	3	5	6	1	1		60	41
   25	Game 23	5	4	5	5	6	4	2	1	3	6		50	41
   26	Game 24	3	5	2	3	2	4	3	2	3	3		30	30
   27	Game 25	5	2	4	2	4	5	2	2	5	2		50	33
   28
   29												mean	37.60	33.68
   30												sd	17.39	5.77

   \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{table}
2. Use the simulation to estimate \(\mathrm { P } ( L > 40 )\) and \(\mathrm { P } ( H > 40 )\).
3. (A) Calculate the exact value of \(\mathrm { P } ( L > 40 )\).
   (B) Comment on how the exact value compares with your estimate of \(\mathrm { P } ( L > 40 )\) in part (v). Hui wonders whether it is appropriate to use the Central Limit Theorem to approximate the distribution of \(X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 }\).
4. (A) State what type of diagram Hui could draw, based on the output from the spreadsheet, to investigate this.
   (B) Explain how she should interpret the diagram.
5. (A) Calculate an approximate value of \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 } > 40 \right)\) using the Central Limit Theorem.
   (B) Comment on how this value compares with your estimate of \(\mathrm { P } ( H > 40 )\) in part (v). \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
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The probability distribution of a random variable \(X\) is shown.

\(x\)	1	3	5	7
P\((X = x)\)	0.4	0.3	0.2	0.1

1. Find E\((X)\) and Var\((X)\). [5]
2. Three independent values of \(X\), denoted by \(X_1\), \(X_2\) and \(X_3\), are chosen. Given that \(X_1 + X_2 + X_3 = 19\), write down all the possible sets of values for \(X_1\), \(X_2\) and \(X_3\) and hence find P\((X_1 = 7)\). [2]
3. 11 independent values of \(X\) are chosen. Use an appropriate formula to find the probability that exactly 4 of these values are 5s. [3]

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

1. Show that \(P(X = 2) = \frac{15}{32}\). [1]

The values of three independent observations of \(X\) are denoted by \(X_1\), \(X_2\) and \(X_3\).

1. Given that \(X_1 + X_2 + X_3 = 9\), determine the probability that at least one of these three values is equal to 2. [6]

Freda chooses values of \(X\) at random until she has obtained \(X = 2\) exactly three times. She then stops.

1. Determine the probability that she chooses exactly 10 values of \(X\). [3]