2.04a Discrete probability distributions

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.

Alisha has four coins. One of these coins is biased so that the probability of obtaining a head is 0.6. The other three coins are fair. Alisha throws the four coins at the same time. The random variable \(X\) denotes the number of heads obtained.

1. Show that the probability of obtaining exactly one head is 0.225. [3]
2. Complete the following probability distribution table for \(X\). [2]

   \(x\)	0	1	2	3	4
   P(\(X = x\))	0.05	0.225			0.075

3. Given that E(\(X\)) = 2.1, find the value of Var(\(X\)). [2]

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 small farm has 5 ducks and 2 geese. Four of these birds are to be chosen at random. The random variable \(X\) represents the number of geese chosen.

1. Draw up the probability distribution of \(X\). [3]
2. Show that \(\mathrm{E}(X) = \frac{8}{7}\) and calculate \(\mathrm{Var}(X)\). [3]
3. When the farmer's dog is let loose, it chases either the ducks with probability \(\frac{3}{5}\) or the geese with probability \(\frac{2}{5}\). If the dog chases the ducks there is a probability of \(\frac{1}{10}\) that they will attack the dog. If the dog chases the geese there is a probability of \(\frac{1}{4}\) that they will attack the dog. Given that the dog is not attacked, find the probability that it was chasing the geese. [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]

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

1. Find the value of \(p\). [2]
2. Find \(\text{E}(S)\). [2]
3. Show that \(\text{E}(S^2) = 9.45\) [2]
4. Find \(\text{Var}(S)\). [2]

Tom and Jess play a game with this spinner. The spinner is spun repeatedly and \(S\) counters are awarded on the outcome of each spin. If \(S\) is even then Tom receives the counters and if \(S\) is odd then Jess receives them. The first player to collect 10 or more counters is the winner.

1. Find the probability that Jess wins after 2 spins. [2]
2. Find the probability that Tom wins after exactly 3 spins. [4]
3. Find the probability that Jess wins after exactly 3 spins. [3]

Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the 5\% level of significance. State your hypotheses clearly. [6]

In a town, 30\% of residents listen to the local radio station. Four residents are chosen at random.

1. State the distribution of the random variable \(X\), the number of these four residents that listen to local radio. [2]
2. On graph paper, draw the probability distribution of \(X\). [3]
3. Write down the most likely number of these four residents that listen to the local radio station. [1]
4. Find E(\(X\)) and Var (\(X\)). [3]

A fair six-sided die is labelled with the numbers 1, 2, 3, 4, 5 and 6. The die is rolled 40 times and the score, \(S\), for each roll is recorded.

1. Find the mean and the variance of \(S\). [2]
2. Find an approximation for the probability that the mean of the 40 scores is less than 3 [3]

Two fair six-sided dice are thrown. The random variable \(X\) denotes the difference between the scores on the two dice. The table shows the probability distribution of \(X\).

\(r\)	0	1	2	3	4	5
P(X = r)	\(\frac{1}{6}\)	\(\frac{5}{18}\)	\(\frac{2}{9}\)	\(\frac{1}{6}\)	\(\frac{1}{9}\)	\(\frac{1}{18}\)

1. Draw a vertical line chart to illustrate the probability distribution. [2]
2. Use a probability argument to show that
   1. P(X = 1) = \(\frac{5}{18}\). [2]
   2. P(X = 0) = \(\frac{1}{6}\). [1]
3. Find the mean value of \(X\). [2]

The probability distribution of the random variable \(X\) is given by the formula $$\mathrm{P}(X = r) = k + 0.01r^2 \text{ for } r = 1, 2, 3, 4, 5.$$

1. Show that \(k = 0.09\). Using this value of \(k\), display the probability distribution of \(X\) in a table. [3]
2. Find \(\mathrm{E}(X)\) and \(\mathrm{Var}(X)\). [5]

A six-sided die is biased such that there is an equal chance of scoring each of the numbers from 1 to 5 but a score of 6 is three times more likely than each of the other numbers.

1. Write down the probability distribution for the random variable, \(X\), the score on a single throw of the die. [4]
2. Show that E\((X) = \frac{33}{8}\). [3]
3. Find E\((4X - 1)\). [2]
4. Find Var\((X)\). [4]

The discrete random variable \(X\) has the probability function shown below. $$P(X = x) = \begin{cases} kx, & x = 2, 3, 4, 5, 6, \\ 0, & \text{otherwise}. \end{cases}$$

1. Find the value of \(k\). [2 marks]
2. Show that E\((X) = \frac{9}{2}\). [3 marks]

Find

1. P\([X > \text{E}(X)]\), [2 marks]
2. E\((2X - 5)\), [2 marks]
3. Var\((X)\). [4 marks]

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]

The table below shows the probability distribution for a discrete random variable \(X\).

\(x\)	0	1	2	3	4 or more
P(X = x)	0.35	0.25	\(k\)	0.14	0.1

Find the value of \(k\). Circle your answer. 0.14 \quad 0.16 \quad 0.18 \quad 1 [1 mark]

A game consists of spinning a circular wheel divided into numbered sectors as shown below. \includegraphics{figure_17} On each spin the score, \(X\), is the value shown in the sector that the arrow points to when the spinner stops. The probability of the arrow pointing at a sector is proportional to the angle subtended at the centre by that sector.

1. Show that \(P(X = 1) = \frac{5}{18}\) [1 mark]
2. Complete the probability distribution for \(X\) in the table below.

   \(x\)	1
   \(P(X = x)\)	\(\frac{5}{18}\)

   [2 marks]

The discrete random variable \(X\) has probability distribution The discrete random variable \(Y\) has probability distribution It is claimed that P(X ≥ 3) is greater than P(Y ≤ 4) Determine if this claim is correct. Fully justify your answer. [4 marks]

The number of flowers which grow on a certain type of plant can be modelled by the discrete random variable \(X\) The probability distribution of \(X\) is given in the table below.

\(x\)	0	1	2	3	4	5 or more
P(\(X = x\))	0.03	0.15	0.22	0.31	0.09	\(p\)

1. Find the value of \(p\) [2 marks]
2. Two plants of this type are randomly selected from a large batch received from a local garden centre. Find the probability that the two plants will produce a total of three flowers. [3 marks]
3. 1. State one assumption necessary for the calculation in part (b) to be valid. [1 mark]
   2. Comment on whether, in reality, this assumption is likely to be valid. [1 mark]

The number of pots of yoghurt, \(X\), consumed per week by adults in Milton is a discrete random variable with probability distribution given by Find \(P(3 \leq X < 6)\) Circle the correct answer. [1 mark] 0.26 \quad\quad 0.31 \quad\quad 0.35 \quad\quad 0.40

The table below shows the probability distribution for a discrete random variable \(X\).

\(x\)	1	2	3	4	5
P(\(X = x\))	\(k\)	\(2k\)	\(4k\)	\(2k\)	\(k\)

Find the value of \(k\). Circle your answer. [1 mark] \(\frac{1}{2}\) \quad \(\frac{1}{4}\) \quad \(\frac{1}{10}\) \quad \(1\)

The discrete random variable \(X\) has the probability function $$P(X = x) = \begin{cases} c(7 - 2x) & x = 0, 1, 2, 3 \\ k & x = 4 \\ 0 & \text{otherwise} \end{cases}$$ where \(c\) and \(k\) are constants.

1. Show that \(16c + k = 1\) [2 marks]
2. Given that \(P(X \geq 3) = \frac{5}{8}\) find the value of \(c\) and the value of \(k\). [2 marks]

Terence owns a local shop. His shop has three checkouts, at least one of which is always staffed. A regular customer observed that the probability distribution for \(N\), the number of checkouts that are staffed at any given time during the spring, is $$P(N = n) = \begin{cases} \frac{3}{4}\left(\frac{1}{4}\right)^{n-1} & \text{for } n = 1, 2 \\ k & \text{for } n = 3 \end{cases}$$

1. Find the value of \(k\). [1 mark]
2. Find the probability that a customer, visiting Terence's shop during the spring, will find at least 2 checkouts staffed. [2 marks]