Construct probability distribution from scenario

1. The random variable \(X\) has probability function

$$\mathrm { P } ( X = x ) = \frac { ( 2 x - 1 ) } { 36 } \quad x = 1,2,3,4,5,6$$

1. Construct a table giving the probability distribution of \(X\). Find
2. \(\mathrm { P } ( 2 < X \leqslant 5 )\),
3. the exact value of \(\mathrm { E } ( X )\).
4. Show that \(\operatorname { Var } ( X ) = 1.97\) to 3 significant figures.
5. Find \(\operatorname { Var } ( 2 - 3 X )\).

6. A fair blue die has faces numbered \(1,1,3,3,5\) and 5 . The random variable \(B\) represents the score when the blue die is rolled.

1. Write down the probability distribution for \(B\).
2. State the name of this probability distribution.
3. Write down the value of \(\mathrm { E } ( B )\). A second die is red and the random variable \(R\) represents the score when the red die is rolled. The probability distribution of \(R\) is

   \(r\)	2	4	6
   \(\mathrm { P } ( R = r )\)	\(\frac { 2 } { 3 }\)	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 6 }\)

4. Find \(\mathrm { E } ( R )\).
5. Find \(\operatorname { Var } ( R )\). Tom invites Avisha to play a game with these dice.
   Tom spins a fair coin with one side labelled 2 and the other side labelled 5 . When Avisha sees the number showing on the coin she then chooses one of the dice and rolls it. If the number showing on the die is greater than the number showing on the coin, Avisha wins, otherwise Tom wins. Avisha chooses the die which gives her the best chance of winning each time Tom spins the coin.
6. Find the probability that Avisha wins the game, stating clearly which die she should use in each case.

1. In a quiz, a team gains 10 points for every question it answers correctly and loses 5 points for every question it does not answer correctly. The probability of answering a question correctly is 0.6 for each question. One round of the quiz consists of 3 questions.

The discrete random variable \(X\) represents the total number of points scored in one round. The table shows the incomplete probability distribution of \(X\)

1. Show that the probability of scoring 15 points in a round is 0.432
2. Find the probability of scoring 0 points in a round.
3. Find the probability of scoring a total of 30 points in 2 rounds.
4. Find \(\mathrm { E } ( X )\)
5. Find \(\operatorname { Var } ( X )\) In a bonus round of 3 questions, a team gains 20 points for every question it answers correctly and loses 5 points for every question it does not answer correctly.
6. Find the expected number of points scored in the bonus round.

2. A fairground game involves trying to hit a moving target with a gunshot. A round consists of up to 3 shots. Ten points are scored if a player hits the target, but the round is over if the player misses. Linda has a constant probability of 0.6 of hitting the target and shots are independent of one another.

1. Find the probability that Linda scores 30 points in a round. The random variable \(X\) is the number of points Linda scores in a round.
2. Find the probability distribution of \(X\).
3. Find the mean and the standard deviation of \(X\). A game consists of 2 rounds.
4. Find the probability that Linda scores more points in round 2 than in round 1.

5. A group of children were each asked to try and complete a task to test hand-eye coordination. Each child repeated the task until he or she had been successful or had made four attempts. The number of attempts made by the children in the group are summarised in the table below.

1. Calculate the mean and standard deviation of the number of attempts made by each child. It is suggested that the number of attempts made by each child could be modelled by a discrete random variable \(X\) with the probability function $$P ( X = x ) = \left\{ \begin{array} { c c } k \left( 20 - x ^ { 2 } \right) , & x = 1,2,3,4 \\ 0 , & \text { otherwise } \end{array} \right.$$
2. Show that \(k = \frac { 1 } { 50 }\).
3. Find \(\mathrm { E } ( X )\).
4. Comment on the suitability of this model.

2 Leila and Caleb are playing a game, using fair six-sided dice and unbiased coins.

• Leila rolls two dice, and her score \(L\) is the total of the scores on the two dice.
• Caleb spins 4 coins and his score \(C\) is three times the number of heads obtained.

The winner of a game is the player with the higher score. If the two scores are equal, the result of the game is a draw. The spreadsheet in Fig. 2 shows a simulation of 20 plays of the game. \begin{table}[h] \end{table}

1. Explain why the value of \(C\) in row 2 is 3 .
2. Use the spreadsheet to estimate \(\mathrm { P } ( C > 6 )\) and \(\mathrm { P } ( L > 6 )\).
3. Use the spreadsheet to estimate the probability that Leila loses a randomly chosen game.
4. Explain why your answers to parts (b) and (c) may not be very close to the true values.
5. Leila claims that the game is fair (that Leila and Caleb each have an equal chance of winning) because both she and Caleb can get a maximum score of 12 and also in the simulation she won exactly \(50 \%\) of the games.
   Make 2 comments about Leila's claim.

6. Penelope makes 8 cakes per week. Each cake costs \(\pounds 20\) to make and sells for \(\pounds 60\). She always sells at least 5 cakes per week. Any cakes left at the end of the week are donated to a food bank. The probability that 5 cakes are sold in a week is \(0 \cdot 3\). She is twice as likely to sell 6 cakes in a week as she is to sell 7 cakes in a week. The expected profit per week is \(\pounds 206\). Construct a probability distribution for the weekly profit.
Additional page, if required. number Write the question number(s) in the left-hand margin. Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}

1. The discrete random variable \(D\) with the following probability distribution represents the score when a 4-sided die is rolled.

1. Write down the name of this distribution. The die is used to play a game and the random variable \(X\) represents the number of points scored. The die is rolled once and if \(D = 2,3\) or 4 then \(X = D\). If \(D = 1\) the die is rolled a second time and \(X = 0\) if \(D = 1\) again, otherwise \(X\) is the sum of the two scores on the die.
2. Show that the probability of scoring 3 points in this game is \(\frac { 5 } { 16 }\)
3. Find the probability of scoring 0 in this game. The table below shows the probability distribution for the remaining values of \(X\).

   \(x\)	0	2	3	4	5
   \(\mathrm { P } ( X = x )\)		\(\frac { 1 } { 4 }\)		\(\frac { 5 } { 16 }\)	\(\frac { 1 } { 16 }\)

4. Find \(\mathrm { E } ( X )\)
5. Find \(\operatorname { Var } ( X )\) The discrete random variable \(R\) represents the number of times the die is rolled in the game.
6. Write down the probability distribution of \(R\). The random variable \(Y = 2 R + 0.5\)
7. Show that \(\mathrm { E } ( Y ) = \mathrm { E } ( X )\) The game is played once.
8. Find \(\mathrm { P } ( X > Y )\)

5. A biased spinner can only land on one of the numbers \(1,2,3\) or 4 . The random variable \(X\) represents the number that the spinner lands on after a single spin and \(\mathrm { P } ( X = r ) = \mathrm { P } ( X = r + 2 )\) for \(r = 1,2\) Given that \(\mathrm { P } ( X = 2 ) = 0.35\)

1. find the complete probability distribution of \(X\). Ambroh spins the spinner 60 times.
2. Find the probability that more than half of the spins land on the number 4 Give your answer to 3 significant figures. The random variable \(Y = \frac { 12 } { X }\)
3. Find \(\mathrm { P } ( Y - X \leqslant 4 )\)

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.

5 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.

3 John plays a game with two unbiased coins. John tosses the coins. If he gets two heads he wins \(\pounds 1\). If he gets two tails he wins 20 p. If he gets one head and one tail he wins nothing. Let \(X\) be the random variable for the amount of money, in pence, John wins per game.

1. Construct a probability distribution table for \(X\).
2. Calculate \(\mathrm { E } ( X )\).
3. John pays \(s\) pence to play the game. State the values of \(s\) for which John should expect to make a loss.

A fair cubical die with faces numbered 1, 1, 1, 2, 3, 4 is thrown and the score noted. The area \(A\) of a square of side equal to the score is calculated, so, for example, when the score on the die is 3, the value of \(A\) is 9.

1. Draw up a table to show the probability distribution of \(A\). [3]
2. Find \(\text{E}(A)\) and \(\text{Var}(A)\). [4]

A box contains 5 discs, numbered 1, 2, 4, 6, 7. William takes 3 discs at random, without replacement, and notes the numbers on the discs.

1. Find the probability that the numbers on the 3 discs are two even numbers and one odd number. [3]

The smallest of the numbers on the 3 discs taken is denoted by the random variable \(S\).

1. By listing all possible selections (126, 246 and so on) draw up the probability distribution table for \(S\). [5]

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 netball team are in a league with three other teams from which one team will progress to the next stage of the competition. The team's coach estimates their chances of winning each of their three matches in the league to be 0.6, 0.5 and 0.3 respectively, and believes these probabilities to be independent of each other.

1. Show that the probability of the team winning exactly two of their three matches is 0.36 [4 marks]

Let the random variable \(W\) be the number of matches that the team win in the league.

1. Find the probability distribution of \(W\). [4 marks]
2. Find E\((W)\) and Var\((W)\). [6 marks]
3. Comment on the coach's assumption that the probabilities of success in each of the three matches are independent. [2 marks]

A game at a school fete is played with a fair coin and a random number generator which generates random integers between 1 and 52 inclusive. It costs 50 pence to play the game. First, the player tosses the coin. If it lands on tails, the player loses. If it lands on heads, the player is allowed to generate a random number. If the number is 1, the player wins £5. If the number is between 2 and 13 inclusive, the player wins £1. If the number is greater than 13, the player loses.

1. Find the probability distribution of the player's profit. [5]
2. Find the mean and standard deviation of the player's profit. [4]
3. Given that 200 people play the game, calculate
   1. the expected number of players who win some money,
   2. the expected profit for the fete. [2]

A game is played as follows. A fair six-sided dice is thrown once. If the score obtained is even, the amount of money, in £, that the contestant wins is half the score on the dice, otherwise it is twice the score on the dice.

1. Find the probability distribution of the amount of money won by the contestant. [3]
2. The contestant pays £5 for every time the dice is thrown. Find the standard deviation of the loss made by the contestant in 120 throws of the dice. [5]

James plays an arcade game. Each time he plays, he puts a £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 £1 coin, James wins the game with a probability of 0.05 and the machine pays out ten £1 coins. The outcomes can be modelled by a random variable \(X\) representing the number of £1 coins gained at the end of a game.

1. Construct a probability distribution table for \(X\). [2]
2. Show that E(\(X\)) = \(-0.25\) and find Var(\(X\)). [4]

James starts off with 10 £1 coins and decides to play exactly 10 games.

1. Find the expected number of £1 coins that James will have at the end of his 10 games. [2]
2. Find the probability that after his 10 games James will have at least 10 £1 coins left. [3]

James plays an arcade game. Each time he plays, he puts a £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 £1 coin, James wins the game with a probability of 0.05 and the machine pays out ten £1 coins. The outcomes can be modelled by a random variable \(X\) representing the number of £1 coins gained at the end of a game.

1. Construct a probability distribution table for \(X\). [2]
2. Show that E(\(X\)) = -0.25 and find Var(\(X\)). [4]

James starts off with 10 £1 coins and decides to play exactly 10 games.

1. Find the expected number of £1 coins that James will have at the end of his 10 games. [2]
2. Find the probability that after his 10 games James will have at least 10 £1 coins left. [3]