Verify probability from combinatorial selection

3 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.

1. Show that the probability that a randomly selected competitor guesses all 4 correctly is \(\frac { 1 } { 35 }\). Let \(X\) represent the number of correct guesses made by a randomly selected competitor. The probability distribution of \(X\) is shown in the table.

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	0	\(\frac { 4 } { 35 }\)	\(\frac { 18 } { 35 }\)	\(\frac { 12 } { 35 }\)	\(\frac { 1 } { 35 }\)

2. Find the expectation and variance of \(X\).

2 In her purse, Katharine has two \(\pounds 5\) notes, two \(\pounds 10\) notes and one \(\pounds 20\) note. She decides to select two of these notes at random to donate to a charity. The total value of these two notes is denoted by the random variable \(\pounds X\).

1. (A) Show that \(\mathrm { P } ( X = 10 ) = 0.1\).
   (B) Show that \(\mathrm { P } ( X = 30 ) = 0.2\). The table shows the probability distribution of \(X\).

   \(r\)	10	15	20	25	30
   \(\mathrm { P } ( X = r )\)	0.1	0.4	0.1	0.2	0.2

2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

4 Martin has won a competition. For his prize he is given six sealed envelopes, of which he is allowed to open exactly three and keep their contents. Three of the envelopes each contain \(\pounds 5\) and the other three each contain \(\pounds 1000\). Since the envelopes are identical on the outside, he chooses three of them at random. Let \(\pounds X\) be the total amount of money that he receives in prize money.

1. Show that \(\mathrm { P } ( X = 15 ) = 0.05\). The probability distribution of \(X\) is given in the table below.

   \(r\)	15	1010	2005	3000
   \(\mathrm { P } ( X = r )\)	0.05	0.45	0.45	0.05

2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

1 Ryan has 6 one-pound coins and 4 two-pound coins. Ryan decides to select 3 of these coins at random to donate to a charity. The total value, in pounds, of these 3 coins is denoted by the random variable \(X\).

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 6 }\). The table below shows the probability distribution of \(X\).

   \(r\)	3	4	5	6
   \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 2 }\)	\(\frac { 3 } { 10 }\)	\(\frac { 1 } { 30 }\)

2. Draw a graph to illustrate the distribution.
3. In this question you must show detailed reasoning. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   Ryan's friend Sasha decides to give the same amount as Ryan does to the charity plus an extra three pounds. The random variable \(Y\) represents the total amount of money, in pounds, given by Ryan and Sasha.
4. Determine each of the following.
   • E (Y)
   • \(\operatorname { Var } ( Y )\)

3 A fair four-sided dice has its faces numbered \(0,1,2,3\). The dice is rolled three times. The discrete random variable \(X\) is the sum of the lowest and highest scores obtained.

1. Show that \(\mathrm { P } ( X = 1 ) = \frac { 3 } { 32 }\). The table below shows the probability distribution of \(X\).

   \(r\)	0	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 64 }\)	\(\frac { 3 } { 32 }\)	\(\frac { 13 } { 64 }\)	\(\frac { 3 } { 8 }\)	\(\frac { 13 } { 64 }\)	\(\frac { 3 } { 32 }\)	\(\frac { 1 } { 64 }\)

2. In this question you must show detailed reasoning. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • The random variable \(Y\) represents the sum of 10 values of \(X\).
     1. State a property of the 10 values of \(X\) that would make it possible to deduce the standard deviation of \(Y\).
     2. Given that this property holds, determine the standard deviation of \(Y\).

1 In a game at a fair, players choose 4 countries from a list of 10 countries. The names of all 10 countries are then put in a box and the player selects 4 of them at random. The random variable \(X\) represents the number of countries that match those which the player originally chose.

1. Show that the probability that a randomly selected player matches all 4 countries is \(\frac { 1 } { 210 }\). Table 1 shows the probability distribution of \(X\). \begin{table}[h]

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 14 }\)	\(\frac { 8 } { 21 }\)	\(\frac { 3 } { 7 }\)	\(\frac { 4 } { 35 }\)	\(\frac { 1 } { 210 }\)

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • A player has to pay \(\pounds 1\) to play the game. The player gets 40 pence back for every country which is matched.
   Find the mean and standard deviation of the player's loss per game.
3. In order to try to attract more customers, the rules will be changed as follows. The game will still cost \(\pounds 1\) to play. The player will get 25 pence back for every country which is matched, plus an additional bonus of \(\pounds 100\) if all four countries are matched. Find the player's mean gain or loss per game with these new rules.

2 In a game at a charity fair, a player rolls 3 unbiased six-sided dice. The random variable \(X\) represents the difference between the highest and lowest scores.

1. Show that \(\mathrm { P } ( X = 0 ) = \frac { 1 } { 36 }\). The table shows the probability distribution of \(X\).

   \(r\)	0	1	2	3	4	5
   \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)	\(\frac { 1 } { 36 }\)	\(\frac { 5 } { 36 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 5 } { 36 }\)

2. Draw a graph to illustrate the distribution.
3. Describe the shape of the distribution.
4. In this question you must show detailed reasoning. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   As a result of playing the game, the player receives \(30 X\) pence from the organiser of the game.
5. Find the variance of the amount that the player receives.
6. The player pays \(k\) pence to play the game. Given that the average profit made by the organiser is 12.5 pence per game, determine the value of \(k\).

3. 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 \(P ( M = 6 ) = \frac { 1 } { 3 }\)
   [0pt] [2] The table shows the probability distribution of \(M\)

   \(m\)	4	5	6	7	8
   \(P ( M = m )\)	\(\frac { 1 } { 15 }\)	\(\frac { 4 } { 15 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 4 } { 15 }\)	\(\frac { 1 } { 15 }\)

   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.
2. Find \(P ( M = S )\)
3. Find \(P ( S = 7 \mid M = S )\)

7. 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 \(P ( M = 6 ) = \frac { 1 } { 3 }\) The table shows the probability distribution of \(M\)

   \(m\)	4	5	6	7	8
   \(P ( M = m )\)	\(\frac { 1 } { 15 }\)	\(\frac { 4 } { 15 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 4 } { 15 }\)	\(\frac { 1 } { 15 }\)

   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.
2. Find \(P ( M = S )\)
3. Find \(P ( S = 7 \mid M = S )\)
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