5.02b Expectation and variance: discrete random variables

7 James throws a discus repeatedly in an attempt to achieve a successful throw. A throw is counted as successful if the distance achieved is over 40 metres. For each throw, the probability that James is successful is \(\frac { 1 } { 4 }\), independently of all other throws. Find the probability that James takes

1. exactly 5 throws to achieve the first successful throw,
2. more than 8 throws to achieve the first successful throw. In order to qualify for a competition, a discus-thrower must throw over 40 metres within at most six attempts. When a successful throw is achieved, no further throws are taken. Find the probability that James qualifies for the competition. Colin is another discus-thrower. For each throw, the probability that he will achieve a throw over 40 metres is \(\frac { 1 } { 3 }\), independently of all other throws. Find the probability that exactly one of James and Colin qualifies for the competition.

4 A fair four-sided spinner has edges numbered 1, 2, 2, 3. A fair three-sided spinner has edges numbered \(- 2 , - 1,1\). Each spinner is spun and the number on the edge on which it comes to rest is noted. The random variable \(X\) is the sum of the two numbers that have been noted.

1. Draw up the probability distribution table for \(X\).
2. Find \(\operatorname { Var } ( X )\).

7 Sharma knows that she has 3 tins of carrots, 2 tins of peas and 2 tins of sweetcorn in her cupboard. All the tins are the same shape and size, but the labels have all been removed, so Sharma does not know what each tin contains. Sharma wants carrots for her meal, and she starts opening the tins one at a time, chosen randomly, until she opens a tin of carrots. The random variable \(X\) is the number of tins that she needs to open.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 6 } { 35 }\).
2. Draw up the probability distribution table for \(X\).
3. Find \(\operatorname { Var } ( X )\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 A fair spinner has sides numbered 1, 2, 2. Another fair spinner has sides numbered \(- 2,0,1\). Each spinner is spun. The number on the side on which a spinner comes to rest is noted. The random variable \(X\) is the sum of the numbers for the two spinners.

1. Draw up the probability distribution table for \(X\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

3 The random variable \(X\) takes the values \(- 2,1,2,3\). It is given that \(\mathrm { P } ( X = x ) = k x ^ { 2 }\), where \(k\) is a constant.

1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

6 Eli has four fair 4 -sided dice with sides labelled \(1,2,3,4\). He throws all four dice at the same time. The random variable \(X\) denotes the number of 2s obtained.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 3 } { 64 }\).
2. Complete the following probability distribution table for \(X\).

   \(x\)	0	1	2	3	4
   \(\mathrm { P } ( X = x )\)	\(\frac { 81 } { 256 }\)			\(\frac { 3 } { 64 }\)	\(\frac { 1 } { 256 }\)

3. Find \(\mathrm { E } ( X )\).
   Eli throws the four dice at the same time on 96 occasions.
4. Use an approximation to find the probability that he obtains at least two 2 s on fewer than 20 of these occasions.

1 The random variable \(X\) takes the values \(- 2,2\) and 3. It is given that $$\mathrm { P } ( X = x ) = k \left( x ^ { 2 } - 1 \right)$$ where \(k\) is a constant.

1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

3 The random variable \(X\) takes the values \(1,2,3,4\). It is given that \(\mathrm { P } ( X = x ) = k x ( x + a )\), where \(k\) and \(a\) are constants.

1. Given that \(\mathrm { P } ( X = 4 ) = 3 \mathrm { P } ( X = 2 )\), find the value of \(a\) and the value of \(k\).
2. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
3. Given that \(\mathrm { E } ( X ) = 3.2\), find \(\operatorname { Var } ( X )\).

6 Harry has three coins:

• One coin is biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 3 }\).
• The second coin is biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 4 }\).
• The third coin is biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 5 }\).

Harry throws the three coins. The random variable \(X\) is the number of heads that he obtains.

1. Draw up the probability distribution table for \(X\).
   Harry has two other coins, each of which is biased so that the probability of obtaining a head when it is thrown is \(p\). He throws all five coins at the same time. The random variable \(Y\) is the number of heads that he obtains.
2. Given that \(\mathrm { P } ( Y = 0 ) = 6 \mathrm { P } ( Y = 5 )\), find the value of \(p\).

5 Jasmine has one \(\\) 5\( coin, two \)\\( 2\) coins and two \(\\) 1\( coins. She selects two of these coins at random. The random variable \)X$ is the total value, in dollars, of these two coins.

1. Show that \(\mathrm { P } ( X = 7 ) = 0.2\).
2. Draw up the probability distribution table for \(X\).
3. Find the value of \(\operatorname { Var } ( X )\).

1 The numbers on the faces of a fair six-sided dice are \(1,2,2,3,3,3\). The random variable \(X\) is the total score when the dice is rolled twice.

1. Draw up the probability distribution table for \(X\).
2. Find the value of \(\operatorname { Var } ( X )\). \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-02_2714_34_143_2012}
3. Find the probability that \(X\) is even given that \(X > 3\).

4 The random variable \(X\) takes the values \(1,2,3,4\) only. The probability that \(X\) takes the value \(x\) is \(k x ( 5 - x )\), where \(k\) is a constant.

1. Draw up the probability distribution table for \(X\), in terms of \(k\).
2. Show that \(\operatorname { Var } ( X ) = 1.05\).

1 A fair red spinner has edges numbered \(1,2,2,3\). A fair blue spinner has edges numbered \(- 3 , - 2 , - 1 , - 1\). Each spinner is spun once and the number on the edge on which each spinner lands is noted. The random variable \(X\) denotes the sum of the resulting two numbers.

1. Draw up the probability distribution table for \(X\).
2. Given that \(\mathrm { E } ( X ) = 0.25\), find the value of \(\operatorname { Var } ( X )\).

6 Three coins \(A , B\) and \(C\) are each thrown once.

• Coins \(A\) and \(B\) are each biased so that the probability of obtaining a head is \(\frac { 2 } { 3 }\).
• Coin \(C\) is biased so that the probability of obtaining a head is \(\frac { 4 } { 5 }\).
  1. Show that the probability of obtaining exactly 2 heads and 1 tail is \(\frac { 4 } { 9 }\).
     

The random variable \(X\) is the number of heads obtained when the three coins are thrown.

Draw up the probability distribution table for \(X\).

Given that \(\mathrm { E } ( X ) = \frac { 32 } { 15 }\), find \(\operatorname { Var } ( X )\).

4 A fair spinner has edges numbered \(0,1,2,2\). Another fair spinner has edges numbered \(- 1,0,1\). Each spinner is spun. The number on the edge on which a spinner comes to rest is noted. The random variable \(X\) is the sum of the numbers for the two spinners.

1. Draw up the probability distribution table for \(X\).
2. Find \(\operatorname { Var } ( X )\).

6 In a game, Jim throws three darts at a board. This is called a 'turn'. The centre of the board is called the bull's-eye. The random variable \(X\) is the number of darts in a turn that hit the bull's-eye. The probability distribution of \(X\) is given in the following table. It is given that \(\mathrm { E } ( X ) = 0.55\).

1. Find the values of \(p\) and \(q\).
2. Find \(\operatorname { Var } ( X )\).
   Jim is practising for a competition and he repeatedly throws three darts at the board.
3. Find the probability that \(X = 1\) in at least 3 of 12 randomly chosen turns.
4. Find the probability that Jim first succeeds in hitting the bull's-eye with all three darts on his 9th turn.

1 The probability distribution table for a random variable \(X\) is shown below. Given that \(\mathrm { E } ( X ) = 0.28\), find the value of \(p\) and the value of \(q\).

1 A competitor in a throwing event has three attempts to throw a ball as far as possible. The random variable \(X\) denotes the number of throws that exceed 30 metres. The probability distribution table for \(X\) is shown below.

1. Given that \(\mathrm { E } ( X ) = 1.1\), find the value of \(p\) and the value of \(r\).
2. Find the numerical value of \(\operatorname { Var } ( X )\).

1 Becky sometimes works in an office and sometimes works at home. The random variable \(X\) denotes the number of days that she works at home in any given week. It is given that $$\mathrm { P } ( X = x ) = k x ( x + 1 )$$ where \(k\) is a constant and \(x = 1,2,3\) or 4 only.

1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

2 The random variable \(X\) takes the values \(- 2 , - 1,0,2,3\). It is given that \(\mathrm { P } ( X = x ) = k \left( x ^ { 2 } + 2 \right)\), where \(k\) is a positive constant.

1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
2. Find the value of \(\operatorname { Var } ( X )\).

3 A fair coin and an ordinary fair six-sided dice are thrown at the same time.The random variable \(X\) is defined as follows.
-If the coin shows a tail,\(X\) is twice the score on the dice.
-If the coin shows a head,\(X\) is the score on the dice if the score is even and \(X\) is 0 otherwise.

1. Draw up the probability distribution table for \(X\) .
2. Find \(\operatorname { Var } ( X )\) .

2 A red fair six-sided dice has faces labelled 1, 1, 1, 2, 2, 2. A blue fair six-sided dice has faces labelled \(1,1,2,2,3,3\). Both dice are thrown. The random variable \(X\) is the product of the scores on the two dice.

1. Draw up the probability distribution table for \(X\).
2. Find \(\mathrm { E } ( X )\).

2 Gohan throws a fair tetrahedral die with faces numbered \(1,2,3,4\). If she throws an even number then her score is the number thrown. If she throws an odd number then she throws again and her score is the sum of both numbers thrown. Let the random variable \(X\) denote Gohan's score.

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 5 } { 16 }\).
2. The table below shows the probability distribution of \(X\).

   \(x\)	2	3	4	5	6	7
   \(\mathrm { P } ( X = x )\)	\(\frac { 5 } { 16 }\)	\(\frac { 1 } { 16 }\)	\(\frac { 3 } { 8 }\)	\(\frac { 1 } { 8 }\)	\(\frac { 1 } { 16 }\)	\(\frac { 1 } { 16 }\)

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

1 The probability distribution of the discrete random variable \(X\) is shown in the table below. Given that \(\mathrm { E } ( X ) = 0.75\), find the values of \(a\) and \(b\).

5 Set \(A\) consists of the ten digits \(0,0,0,0,0,0,2,2,2,4\).
Set \(B\) consists of the seven digits \(0,0,0,0,2,2,2\).
One digit is chosen at random from each set. The random variable \(X\) is defined as the sum of these two digits.

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 3 } { 7 }\).
2. Tabulate the probability distribution of \(X\).
3. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
4. Given that \(X = 2\), find the probability that the digit chosen from set \(A\) was 2 .