2.04a Discrete probability distributions

1. A biased 4 -sided spinner has the numbers \(6,7,8\) and 10 on it.

The discrete random variable \(X\) represents the score when the spinner is spun once and has the following probability distribution, where \(q\) is a probability.

1. Find the value of \(q\) Karen spins the spinner repeatedly until she either gets a 7 or she has taken 4 spins.
2. Show that the probability that Karen stops after taking her 3rd spin is 0.128 The random variable \(S\) represents the number of spins Karen takes.
3. Find the probability distribution for \(S\) The random variable \(N\) represents the number of times Karen gets a 7
4. Find \(\mathrm { P } ( S > N )\)

1. The discrete random variable \(D\) has the following probability distribution

where \(k\) is a constant.

1. Show that the value of \(k\) is \(\frac { 600 } { 137 }\) The random variables \(D _ { 1 }\) and \(D _ { 2 }\) are independent and each have the same distribution as \(D\).
2. Find \(\mathrm { P } \left( D _ { 1 } + D _ { 2 } = 80 \right)\) Give your answer to 3 significant figures. A single observation of \(D\) is made.
   The value obtained, \(d\), is the common difference of an arithmetic sequence.
   The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral \(Q\)
3. Find the exact probability that the smallest angle of \(Q\) is more than \(50 ^ { \circ }\)

1. The discrete random variable \(X\) has the following probability distribution

where

• \(\quad a , b\) and \(c\) are distinct integers \(( a < b < c )\)
• all the probabilities are greater than zero
  1. Find
     1. the value of a
     2. the value of \(b\)
     3. the value of \(c\)

Show your working clearly. The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\)

Find \(\mathrm { P } \left( X _ { 1 } = X _ { 2 } \right)\) \section*{Question 6 continued.} \section*{Question 6 continued.}

13

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

11 Alex models the number of goals that a local team will score in any match as follows. The number of goals scored in any match is independent of the number of goals scored in any other match.

1. Alex chooses 3 matches at random. Use the model to determine the probability of each of the following.
   1. The team will score a total of exactly 1 goal in the 3 matches.
   2. The numbers of goals scored in the first 2 of the 3 matches will be equal, but the number of goals scored in the 3rd match will be different. During the first 10 matches this season, the team scores a total of 31 goals.
2. Without carrying out a formal test, explain briefly whether this casts doubt on the validity of Alex's model. \section*{END OF QUESTION PAPER}

1 The probability distribution for the discrete random variable \(X\) is shown below. Find the value of \(a\).

9 The table shows the probability distribution for the discrete random variable \(X\). You are given that \(q\) is a positive constant.

1. Determine the value of \(q\).
2. Calculate \(\mathrm { P } ( X \leqslant 4 )\). Two independent values of \(X\) are taken.
3. Determine the probability that the sum of the two values is 3 . Fifty independent values of \(X\) are taken.
4. Find the probability that a value of 2 occurs exactly 17 times.

6 The probability distribution for the discrete random variable \(X\) is shown below.

1. Determine the value of \(p\).
2. Determine the modal value of \(X\).

4 A biased octagonal dice has faces numbered from 1 to 8 . The discrete random variable \(X\) is the score obtained when the dice is rolled once. The probability distribution of \(X\) is shown in the table below.

1. Determine the value of \(p\).
2. Find the probability that a score of at least 4 is obtained when the dice is rolled once. The dice is rolled 30 times.
3. Determine the probability that a score of 8 occurs exactly twice.

6 The probability distribution of the discrete random variable \(X\) is shown in the table.

1. Calculate the value of the constant \(a\).
2. A single value of \(X\) is chosen at random. Find the probability that the value is an odd number.
3. Two independent values of \(X\) are chosen at random. Calculate the probability that the total of the two values is 3 .

15

1. Show that \(\sum _ { r = 1 } ^ { \infty } 0.99 ^ { r - 1 } \times 0.01 = 1\). Kofi is a very good table tennis player. Layla is determined to beat him.
   Every week they play one match of table tennis against each other. They will stop playing when Layla wins the match for the first time. \(X\) is the discrete random variable "the number of matches they play in total". Kofi models the situation using the probability function \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = 0.99 ^ { \mathrm { r } - 1 } \times 0.01 \quad r = 1,2,3,4 , \ldots\) Kofi states that he is \(95 \%\) certain that Layla will not beat him within 6 years.
2. Determine whether Kofi's statement is consistent with his model. In between matches, Layla practises, but Kofi does not.
3. Explain why Layla might disagree with Kofi's model. Layla models the situation using the probability function \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { kr } ^ { 2 } \quad r = 1,2,3,4,5,6,7,8\).
4. Explain how Layla's model takes into account the fact that she practises between matches, but Kofi's does not. Layla states that she is \(95 \%\) certain that she will beat Kofi within the first 6 matches.
5. Determine whether Layla's statement is consistent with her model.

5. A biased tetrahedral die has faces numbered \(0,1,2\) and 3 . The die is rolled and the number face down on the die, \(X\), is recorded. The probability distribution of \(X\) is If \(X = 3\) then the final score is 3
If \(X \neq 3\) then the die is rolled again and the final score is the sum of the two numbers. The random variable \(T\) is the final score.

1. Find \(\mathrm { P } ( T = 2 )\)
2. Find \(\mathrm { P } ( T = 3 )\)
3. Given that the die is rolled twice, find the probability that the final score is 3

1. The discrete random variable \(X\) represents the score when a biased spinner is spun. The probability distribution of \(X\) is given by

where \(p\) and \(q\) are probabilities.

1. Find \(\mathrm { E } ( X )\). Given that \(\operatorname { Var } ( X ) = 2.5\)
2. find the value of \(p\).
3. Hence find the value of \(q\). Amar is invited to play a game with the spinner.
   The spinner is spun once and \(X _ { 1 }\) is the score on the spinner. If \(X _ { 1 } > 0\) Amar wins the game.
   If \(X _ { 1 } = 0\) Amar loses the game.
   If \(X _ { 1 } < 0\) the spinner is spun again and \(X _ { 2 }\) is the score on this second spin and if \(X _ { 1 } + X _ { 2 } > 0\) Amar wins the game, otherwise Amar loses the game.
4. Find the probability that Amar wins the game. Amar does not want to lose the game.
   He says that because \(\mathrm { E } ( X ) > 0\) he will play the game.
5. State, giving a reason, whether or not you would agree with Amar.

1. The discrete random variable \(X\) has the following probability distribution

Given that \(\mathrm { E } ( X ) = 0.5\)

1. find the value of \(a\). Given also that \(\operatorname { Var } ( X ) = 5.01\)
2. find the value of \(b\) and the value of \(c\). The random variable \(Y = 5 - 8 X\)
3. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\)
4. Find \(\mathrm { P } \left( 4 X ^ { 2 } > Y \right)\)

1. A red spinner is designed so that the score \(R\) is given by the following probability distribution.

1. Show that \(\mathrm { E } \left( R ^ { 2 } \right) = 15.8\) Given also that \(\mathrm { E } ( R ) = 3.7\)
2. find the standard deviation of \(R\), giving your answer to 2 decimal places. A yellow spinner is designed so that the score \(Y\) is given by the probability distribution in the table below. The cumulative distribution function \(\mathrm { F } ( y )\) is also given.

   \(y\)	2	3	4	5	6
   \(\mathrm { P } ( Y = y )\)	0.1	0.2	0.1	\(a\)	\(b\)
   \(\mathrm {~F} ( y )\)	0.1	0.3	0.4	\(c\)	\(d\)

3. Write down the value of \(d\) Given that \(\mathrm { E } ( Y ) = 4.55\)
4. find the value of \(c\) Pabel and Jessie play a game with these two spinners.
   Pabel uses the red spinner.
   Jessie uses the yellow spinner.
   They take turns to spin their spinner.
   The winner is the first person whose spinner lands on the number 2 and the game ends. Jessie spins her spinner first.
5. Find the probability that Jessie wins on her second spin.
6. Calculate the probability that, in a game, the score on Pabel's first spin is the same as the score on Jessie's first spin.

2. A spinner can land on the numbers \(2,4,5,7\) or 8 only. The random variable \(X\) represents the number that this spinner lands on when it is spun once. The probability distribution of \(X\) is given in the table below.

1. Find \(\mathrm { P } ( 2 X - 3 > 5 )\) Given that \(\mathrm { E } ( X ) = 4.6\)
2. show that \(\operatorname { Var } ( X ) = 4.14\) The random variable \(Y = a X - b\) where \(a\) and \(b\) are positive constants.
   Given that $$\mathrm { E } ( Y ) = 13.4 \quad \text { and } \quad \operatorname { Var } ( Y ) = 66.24$$
3. find the value of \(a\) and the value of \(b\) In a game Sam and Alex each spin the spinner once, landing on \(X _ { 1 }\) and \(X _ { 2 }\) respectively.
   Sam's score is given by the random variable \(S = X _ { 1 }\) Alex's score is given by the random variable \(R = 2 X _ { 2 } - 3\) The person with the higher score wins the game. If the scores are the same it is a draw.
4. Find the probability that Sam wins the game.

1. The discrete random variable \(X\) has probability distribution

where \(a\) and \(b\) are constants.

1. Write down an equation for \(a\) and \(b\).
2. Calculate \(\mathrm { E } ( X )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 3.5\)
3. 1. find a second equation in \(a\) and \(b\),
   2. hence find the value of \(a\) and the value of \(b\).
4. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 5 - 3 X\)
5. Find \(\mathrm { P } ( Y > 0 )\).
6. Find
   1. \(\mathrm { E } ( Y )\),
   2. \(\operatorname { Var } ( Y )\).

1. A biased tetrahedral die has faces numbered \(0,1,2\) and 3 . The die is rolled and the number face down on the die, \(X\), is recorded. The probability distribution of \(X\) is

If \(X = 3\) then the final score is 3
If \(X \neq 3\) then the die is rolled again and the final score is the sum of the two numbers.
The random variable \(T\) is the final score.

1. Find \(\mathrm { P } ( T = 2 )\)
2. Find \(\mathrm { P } ( T = 3 )\)
3. Given that the die is rolled twice, find the probability that the final score is 3 \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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1. The discrete random variable \(X\) has probability distribution given by

where \(a\) is a constant.

1. Find the value of \(a\).
2. Write down \(\mathrm { E } ( X )\).
3. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 6 - 2 X\)
4. Find \(\operatorname { Var } ( Y )\).
5. Calculate \(\mathrm { P } ( X \geqslant Y )\).

3. A fair six-sided die is rolled. The random variable \(Y\) represents the score on the uppermost, face.

1. Write down the probability function of \(Y\).
2. State the name of the distribution of \(Y\). Find the value of
3. \(\mathrm { E } ( 6 Y + 2 )\),
4. \(\operatorname { Var } ( 4 Y - 2 )\).

5. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) , & x = 0,1,2 \\ k ( x - 2 ) , & x = 3 \\ 0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Show that \(k = 0.25\).
2. Find \(\mathrm { E } ( X )\) and show that \(\mathrm { E } \left( X ^ { 2 } \right) = 2.5\).
3. Find \(\operatorname { Var } ( 3 X - 2 )\). Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
4. Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0\).
5. Find the complete probability function for \(X _ { 1 } + X _ { 2 }\).
6. Find \(\mathrm { P } \left( 1.3 \leq X _ { 1 } + X _ { 2 } \leq 3.2 \right)\).

4. The random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = k x , \quad x = 1,2 , \ldots , 5$$

1. Show that \(k = \frac { 1 } { 15 }\). Find
2. \(\mathrm { P } ( X < 4 )\),
3. \(\mathrm { E } ( X )\),
4. \(\mathrm { E } ( 3 X - 4 )\).

6. A discrete random variable is such that each of its values is assumed to be equally likely.

1. Write down the name of the distribution that could be used to model this random variable.
2. Give an example of such a distribution.
3. Comment on the assumption that each value is equally likely.
4. Suggest how you might refine the model in part (a).

2. The random variable \(X\) has probability distribution

1. Given that \(\mathrm { E } ( X ) = 3.5\), write down two equations involving \(p\) and \(q\). Find
2. the value of \(p\) and the value of \(q\),
3. \(\operatorname { Var } ( X )\),
4. \(\operatorname { Var } ( 3 - 2 X )\).

5. (a) Write down two reasons for using statistical models.
(b) Give an example of a random variable that could be modelled by

1. a normal distribution,
2. a discrete uniform distribution.