2.04a Discrete probability distributions

The probability distribution of a random variable \(X\) is given in the table.

1. Find the value of \(p\). [2]
2. Two values of \(X\) are chosen at random. Find the probability that the product of these values is 0. [3]

The probability distribution of the discrete random variable \(X\) is given in Fig. 4.

\(r\)	0	1	2	3	4
P\((X = r)\)	0.2	0.15	0.3	\(k\)	0.25

Fig. 4

1. Find the value of \(k\). [2]

\(X_1\) and \(X_2\) are two independent values of \(X\).

1. Find P\((X_1 + X_2 = 6)\). [3]

A die in the form of a dodecahedron has its faces numbered from 1 to 12. The die is biased so that the probability that a score of 12 is achieved is different from any other score. The probability distribution of \(X\), the score on the die, is given in the table in terms of \(p\) and \(k\), where \(0 < p < 1\) and \(k\) is a positive integer. Sam rolls the die 30 times, Leo rolls the die 60 times and Nina rolls the die 120 times. They each plot their scores on bar line graphs.

1. Explain whose graph is most likely to give the best representation of the theoretical probability distribution for the score on the die. [1]
2. Find \(p\) in terms of \(k\). [2]
3. Determine, in terms of \(k\), the expected number of times Nina rolls a 12. [3]
4. Given that Nina rolls a 12 on 32 occasions, calculate an estimate of the value of \(k\). [2]

Nina rolls the die a further 30 times.

1. Use your answer to part (d) to calculate an estimate for the probability that she obtains a 12 exactly 8 times in these 30 rolls. [2]

Antonio arrives at a train station at a random point in time. The trains to his desired destination are scheduled to depart at 12-minute intervals.

1. Assume that Antonio gets on the next train.
   1. Suggest an appropriate distribution to model his waiting time and give the parameters.
   2. State the mean and the variance of this distribution.
   3. State an assumption you have made in suggesting this distribution. [4]
2. Now assume that the probability that Antonio misses the next available train because he is distracted by his smartphone is \(0 \cdot 12\). If he misses the next available train, he is sure to get on the one after that.
   1. Find the probability that he waits between 9 and 19 minutes.
   2. Given that he waits between 9 and 19 minutes, find the probability that he gets on the first train. [6]

The probability distribution of a random variable \(X\) is modelled as follows. $$\text{P}(X = x) = \begin{cases} \frac{k}{x} & x = 1, 2, 3, 4, \\ 0 & \text{otherwise,} \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{12}{25}\). [2]
2. Show in a table the values of \(X\) and their probabilities. [1]
3. The values of three independent observations of \(X\) are denoted by \(X_1\), \(X_2\) and \(X_3\). Find P\((X_1 > X_2 + X_3)\). [3]

In a game, a player notes the values of successive independent observations of \(X\) and keeps a running total. The aim of the game is to reach a total of exactly 7.

1. Determine the probability that a total of exactly 7 is first reached on the 5th observation. [5]

Miguel has six numbered tiles, labelled 2, 2, 3, 3, 4, 4. He selects two tiles at random, without replacement. The variable \(M\) denotes the sum of the numbers on the two tiles.

1. Show that \(P(M = 6) = \frac{1}{3}\) [2]

The table shows the probability distribution of \(M\) Miguel returns the two tiles to the collection. Now Sofia selects two tiles at random from the six tiles, without replacement. The variable \(S\) denotes the sum of the numbers on the two tiles that Sofia selects.

1. Find \(P(M = S)\) [3]
2. Find \(P(S = 7 | M = S)\) [4]

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]