5.02a Discrete probability distributions: general

7 On a multiple choice examination paper, each question has five alternative answers given, only one of which is correct. For each question, candidates gain 4 marks for a correct answer but lose 1 mark for an incorrect answer.

1. James guesses the answer to each question.
   1. Copy and complete the following table for the probability distribution of \(X\), the number of marks obtained by James for each question.

      \(\boldsymbol { x }\)	4	- 1
      \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)

   2. Hence find \(\mathrm { E } ( X )\).
2. Karen is able to eliminate two of the incorrect answers from the five alternative answers given for each question before guessing the answer from those remaining. Given that the examination paper contains 24 questions, calculate Karen's expected total mark.

1 The discrete random variable \(X\) has the following probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} \frac { 5 - x } { 10 } & x = 1,2,3,4 \\ 0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X \geq 3 )\) Circle your answer.
0.1
0.15
0.2
0.3

1 The discrete random variable \(X\) has the following probability distribution Find \(\mathrm { P } ( X > 18 )\) Circle your answer.
0.1
0.2
0.7
0.8

3 The discrete random variable \(A\) has the following probability distribution function $$\mathrm { P } ( A = a ) = \begin{cases} 0.45 & a = 0 \\ 0.25 & a = 1 \\ 0.3 & a = 2 \\ 0 & \text { otherwise } \end{cases}$$ 3

1. Find the median of \(A\) 3
2. Find the standard deviation of \(A\), giving your answer to three significant figures.
   3
3. \(\quad\) Find \(\operatorname { Var } ( 9 A - 2 )\)

1 The discrete random variable \(X\) has probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} 0.45 & x = 1 \\ 0.25 & x = 2 \\ 0.25 & x = 3 \\ 0.05 & x = 4 \\ 0 & \text { otherwise } \end{cases}$$ State the mode of \(X\) Circle your answer.
0.25
0.45
1
2.5

4 The discrete random variable \(X\) follows a discrete uniform distribution and takes values \(1,2,3 , \ldots , n\). The discrete random variable \(Y\) is defined by \(Y = 2 X\) 4

1. Using the standard results for \(\sum n , \sum n ^ { 2 }\) and \(\operatorname { Var } ( a X + b )\), prove that $$\operatorname { Var } ( Y ) = \frac { n ^ { 2 } - 1 } { 3 }$$ 4
2. A spinning toy can land on one of four values: 2, 4, 6 or 8
   Using a discrete uniform distribution, find the probability that the next value the toy lands on is greater than 2 4
3. State an assumption that is required for the discrete uniform distribution used in part (b) to be valid.

6 The discrete random variable \(X\) has probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} a & x = 0 \\ b & x = 1 \\ c & x = 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a , b\) and \(c\) are constants.
The mean of \(X\) is 1.2 and the variance of \(X\) is 0.56
6

1. Deduce the values of \(a , b\) and \(c\) 6
2. The continuous random variable \(Y\) is independent of \(X\) and has variance 15 Find \(\operatorname { Var } ( X - 2 Y - 11 )\) [0pt] [2 marks]

1 The discrete random variable \(A\) takes only the values 0,2 and 4, and has cumulative distribution function \(\mathrm { F } ( a ) = \mathrm { P } ( A \leq a )\) Find \(\mathrm { P } ( A = 2 )\) Circle your answer. \(0 \quad 0.4 \quad 0.6 \quad 0.8\)

1 In a promotion for a new type of cereal, a toy dinosaur is included in each pack. There are three different types of dinosaur to collect. They are distributed, with equal probability, randomly and independently in the packs. Sam is trying to collect all three of the dinosaurs.

1. Find the probability that Sam has to open only 3 packs in order to collect all three dinosaurs. Sam continues to open packs until she has collected all three dinosaurs, but once she has opened 6 packs she gives up even if she has not found all three. The random variable \(X\) represents the number of packs which Sam opens.
2. Complete the table below, using the copy in the Printed Answer Booklet, to show the probability distribution of \(X\).

   \(r\)	3	4	5	6
   \(\mathrm { P } ( X = r )\)		\(\frac { 2 } { 9 }\)	\(\frac { 14 } { 81 }\)

   \section*{(iii) In this question you must show detailed reasoning.} Find
   • \(\mathrm { E } ( X )\) and
   • \(\operatorname { Var } ( X )\).

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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1. Two boxes, A and B , each contain a large number of coins.

In box A

• there are only 1 p coins and 2 p coins
• the ratio of 1 p coins to 2 p coins is \(1 : 3\)

In box B

• there are only 2 p coins and 5 p coins
• the ratio of 2 p coins to 5 p coins is \(1 : 4\)

One coin is randomly selected from box A and two coins are randomly selected from box B The random variable \(T\) represents the total of the values of the three coins selected.

1. Find the sampling distribution of \(T\) The random variable \(M\) represents the median of the values of the three coins selected.
2. Find the sampling distribution of \(M\)

6 James plays an arcade game. Each time he plays, he puts a \(\pounds 1\) coin in the slot to start the game. The possible outcomes of each game are as follows: James loses the game with a probability of 0.7 and the machine pays out nothing, James draws the game with a probability of 0.25 and the machine pays out a \(\pounds 1\) coin, James wins the game with a probability of 0.05 and the machine pays out ten \(\pounds 1\) coins. The outcomes can be modelled by a random variable \(X\) representing the number of \(\pounds 1\) coins gained at the end of a game.

1. Construct a probability distribution table for \(X\).
2. Show that \(\mathrm { E } ( X ) = - 0.25\) and find \(\operatorname { Var } ( X )\). James starts off with \(10 \pounds 1\) coins and decides to play exactly 10 games.
3. Find the expected number of \(\pounds 1\) coins that James will have at the end of his 10 games.
4. Find the probability that after his 10 games James will have at least \(10 \pounds 1\) coins left.

1 The discrete random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$\mathrm { G } _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 } ,$$ where \(a\) is a constant.

1. Find, in either order, the value of \(a\) and the set of values that \(X\) can take.
2. Find the value of \(\mathrm { E } ( X )\).

2 A discrete random variable \(X\) has the following probability distribution.

1. Write down the probability generating function of \(X\).
2. \(T\) is the sum of ten independent observations of \(X\). Use the probability generating function of \(T\) to find
   1. \(\mathrm { E } ( T )\),
   2. \(\mathrm { P } ( T = 8 )\).

3 The probability distribution of the discrete random variable \(X\) is defined as follows. $$\mathrm { P } ( X = x ) = k ( 2 + x ) ( 5 - x ) \quad \text { for } x = 0,1,2,3,4$$

1. Show that \(k = \frac { 1 } { 50 }\).
2. Find the variance of \(X\).
3. Find \(\mathrm { P } ( X = 4 \mid X > 0 )\).

1 The discrete random variable X has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$G _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 } ,$$ where \(a\) is a constant.

1. Find, in either order, the value of \(a\) and the set of values that \(X\) can take.
2. Find the value of \(\mathrm { E } ( X )\).

1 The discrete random variable X has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$G _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 }$$ where \(a\) is a constant.

1. Find, in either order, the value of \(a\) and the set of values that \(X\) can take.
2. Find the value of \(\mathrm { E } ( X )\).

12 A faulty random number generator generates odd digits according to the probability distribution for the random variable \(X\) given in the following table.

1. Find \(p\).
2. Find \(\mathrm { E } ( X )\) and \(\mathrm { E } \left( X ^ { 2 } \right)\).
3. Deduce the value of \(\operatorname { Var } ( X )\). A second random number generator generates odd digits each with equal probability. Both random generators are operated once.
4. Find the probability that both generate a prime number.
5. Given that the first generates 1, 3 or 5, find the probability that both generate a power of 3 . 1315 pupils, including two sisters, are placed in random order in a line.
6. What is the probability that the sisters are not next to each other?
7. How many arrangements are there with 9 pupils between the sisters? A team of 5 is chosen from the 15 pupils.
8. How many ways are there of choosing the team if no more than one of the sisters can be in the team? Having chosen the first team, a second team of 5 pupils is chosen from the remaining 10 pupils.
9. How many ways are there of choosing the teams if each sister is in one or other of the teams?

12 A game is played in which the number of points scored, \(X\), has the probability distribution given in the following table.

1. Write down the probability generating function of \(X\).
2. Use this generating function to find the mean and variance of \(X\).
3. The game is played 4 times (independently) and the total number of points scored is denoted by \(Y\). Show that the probability generating function of \(Y\) can be written in the form $$\frac { \left( a + t ^ { 2 } \right) ^ { 8 } } { b t ^ { 4 } }$$ where \(a\) and \(b\) are constants.
4. Hence find \(\mathrm { P } ( Y < 0 )\).

Sharik attempts a multiple choice revision question on-line. There are 3 suggested answers, one of which is correct. When Sharik chooses an answer the computer indicates whether the answer is right or wrong. Sharik first chooses one of the three suggested answers at random. If this answer is wrong he has a second try, choosing an answer at random from the remaining 2. If this answer is also wrong Sharik then chooses the remaining answer, which must be correct.

1. Draw a fully labelled tree diagram to illustrate the various choices that Sharik can make until the computer indicates that he has answered the question correctly. [4]
2. The random variable \(X\) is the number of attempts that Sharik makes up to and including the one that the computer indicates is correct. Draw up the probability distribution table for \(X\) and find E\((X)\). [4]

A discrete random variable \(Y\) has probability function $$\mathrm{P}(Y = y) = \begin{cases} k(3 - y) & y = 1, 2 \\ k(y^2 - 8) & y = 3, 4, 5 \\ k & y = 6 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{1}{30}\) [2]

Find the exact value of

1. P\((1 < Y \leqslant 4)\) [2]
2. E\((Y)\) [2]

The random variable \(X = 15 - 2Y\)

1. Calculate P\((Y \geqslant X)\) [3]
2. Calculate Var\((X)\) [4]

A discrete random variable \(X\) has the probability function shown in the table below.

1. Given that E(\(X\)) = \(\frac{2}{3}\), find \(a\). [3]
2. Find the exact value of Var (\(X\)). [3]
3. Find the exact value of P(\(X \leq 15\)). [1]

The probability function of a discrete random variable \(X\) is given by $$p(x) = kx^2 \quad x = 1, 2, 3$$ where \(k\) is a positive constant.

1. Show that \(k = \frac{1}{14}\) [2]

Find

1. P\((X \geq 2)\) [2]
2. E\((X)\) [2]
3. Var\((1-X)\) [4]

The discrete random variable \(Y\) has probability distribution where \(a\), \(b\) and \(c\) are constants. The cumulative distribution function F(\(y\)) of \(Y\) is given in the following table where \(d\) is a constant.

1. Find the value of \(a\), the value of \(b\), the value of \(c\) and the value of \(d\). [5]
2. Find \(\text{P}(3Y + 2 \geq 8)\). [2]