Two unknowns from sum and expectation

2 The random variable \(X\) can take only the values \(- 2 , - 1,0,1,2\). The probability distribution of \(X\) is given in the following table. Given that \(\mathrm { P } ( X \geqslant 0 ) = 3 \mathrm { P } ( X < 0 )\), find the values of \(p\) and \(q\).

4 Jacob has four coins. One of the coins is biased such that when it is thrown the probability of obtaining a head is \(\frac { 7 } { 10 }\). The other three coins are fair. Jacob throws all four coins once. The number of heads that he obtains is denoted by the random variable \(X\). The probability distribution table for \(X\) is as follows.

1. Show that \(a = \frac { 1 } { 5 }\) and find the values of \(b\) and \(c\).
2. Find \(\mathrm { E } ( X )\).
   Jacob throws all four coins together 10 times.
3. Find the probability that he obtains exactly one head on fewer than 3 occasions.
4. Find the probability that Jacob obtains exactly one head for the first time on the 7th or 8th time that he throws the 4 coins.

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 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 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 \(b\).

1 The discrete random variable \(X\) has the following probability distribution.

1. Write down an equation satisfied by \(a\) and \(b\).
2. Given that \(\mathrm { E } ( X ) = 4\), find \(a\) and \(b\).

2 The probability distribution of the random variable \(X\) is shown in the following table. The mean of \(X\) is 1.05 .

1. Write down two equations involving \(p\) and \(q\) and hence find the values of \(p\) and \(q\).
2. Find the variance of \(X\).

1 The discrete random variable \(X\) has the following probability distribution. Given that \(\mathrm { E } ( X ) = 3.05\), find the values of \(p\) and \(q\).

5 The probability distribution of a discrete random variable, \(X\), is given in the table. It is given that the expectation, \(\mathrm { E } ( X )\), is \(1 \frac { 1 } { 4 }\).

1. Calculate the values of \(p\) and \(q\).
2. Calculate the standard deviation of \(X\).

3 In a game of darts, a player throws three darts. Let \(X\) represent the number of darts which hit the bull's-eye. The probability distribution of \(X\) is shown in the table.

1. (A) Show that \(p + q = 0.15\).
   (B) Given that the expectation of \(X\) is 0.67 , show that \(2 p + 3 q = 0.32\).
   (C) Find the values of \(p\) and \(q\).
2. Find the variance of \(X\).

2 In a game of darts, a player throws three darts. Let \(X\) represent the number of darts which hit the bull's-eye. The probability distribution of \(X\) is shown in the table.

1. (A) Show that \(p + q = 0.15\).
   (B) Given that the expectation of \(X\) is 0.67 , show that \(2 p + 3 q = 0.32\).
   (C) Find the values of \(p\) and \(q\).
2. Find the variance of \(X\).

4. A discrete random variable \(X\) has the probability distribution given in the table below, where \(a\) and \(b\) are constants. Given \(\mathrm { E } ( X ) = \frac { 9 } { 5 }\)

1. 1. find two simultaneous equations for \(a\) and \(b\),
   2. show that \(a = \frac { 1 } { 20 }\) and find the value of \(b\).
2. Specify the cumulative distribution function \(\mathrm { F } ( x )\) for \(x = - 1,0,1\), 2 and 3
3. Find \(\mathrm { P } ( X < 2.5 )\).
4. Find \(\operatorname { Var } ( 3 - 2 X )\).
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2. The discrete random variable \(X\) has the following probability distribution.

1. Find \(\mathrm { E } ( X )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 4.54\)
2. find the value of \(a\) and the value of \(b\). The random variable \(Y = 3 - 2 X\)
3. Find \(\operatorname { Var } ( Y )\).

9 The random variable \(X\) has probability distribution given by $$\mathrm { P } ( X = x ) = a + b x \quad \text { for } x = 1,2 \text { and } 3 ,$$ where \(a\) and \(b\) are constants.

1. Show that \(3 a + 6 b = 1\).
2. Given that \(\mathrm { E } ( X ) = \frac { 5 } { 3 }\), find \(a\) and \(b\).

1. Tisam is playing a game.

She uses a ball, a cup and a spinner.
The random variable \(X\) represents the number the spinner lands on when it is spun. The probability distribution of \(X\) is given in the following table where \(a , b , c\) and \(d\) are probabilities.
To play the game

• the spinner is spun to obtain a value of \(x\)
• Tisam then stands \(x \mathrm {~cm}\) from the cup and tries to throw the ball into the cup

The event \(S\) represents the event that Tisam successfully throws the ball into the cup.
To model this game Tisam assumes that

• \(\mathrm { P } ( S \mid \{ X = x \} ) = \frac { k } { x }\) where \(k\) is a constant
• \(\mathrm { P } ( S \cap \{ X = x \} )\) should be the same whatever value of \(x\) is obtained from the spinner

Using Tisam's model,

1. show that \(c = \frac { 8 } { 5 } b\)
2. find the probability distribution of \(X\) Nav tries, a large number of times, to throw the ball into the cup from a distance of 100 cm .
   He successfully gets the ball in the cup \(30 \%\) of the time.
3. State, giving a reason, why Tisam's model of this game is not suitable to describe Nav playing the game for all values of \(X\)

11 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.
2. Find \(p\) in terms of \(k\).
3. Determine, in terms of \(k\), the expected number of times Nina rolls a 12 .
4. Given that Nina rolls a 12 on 32 occasions, calculate an estimate of the value of \(k\). Nina rolls the die a further 30 times.
5. 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.

4 A discrete random variable \(W\) has the probability distribution shown in the following table, in which \(a\) and \(b\) are constants. It is given that \(\mathrm { E } ( W - 60 ) = 0.15\). Determine the value of \(\operatorname { Var } ( 4 W - 60 )\).

2 The probability distribution of a discrete random variable \(W\) is given in the table. Given that \(\mathrm { E } ( W ) = 1.61\), find the value of \(\operatorname { Var } ( 3 W + 2 )\).

1. The discrete random variable \(X\) represents the score when a biased spinner is spun. The probability distribution of \(X\) is given by

where \(p\) and \(q\) are probabilities.

1. Find \(\mathrm { E } ( X )\). Given that \(\operatorname { Var } ( X ) = 2.5\)
2. find the value of \(p\).
3. Hence find the value of \(q\). Amar is invited to play a game with the spinner.
   The spinner is spun once and \(X _ { 1 }\) is the score on the spinner. If \(X _ { 1 } > 0\) Amar wins the game.
   If \(X _ { 1 } = 0\) Amar loses the game.
   If \(X _ { 1 } < 0\) the spinner is spun again and \(X _ { 2 }\) is the score on this second spin and if \(X _ { 1 } + X _ { 2 } > 0\) Amar wins the game, otherwise Amar loses the game.
4. Find the probability that Amar wins the game. Amar does not want to lose the game.
   He says that because \(\mathrm { E } ( X ) > 0\) he will play the game.
5. State, giving a reason, whether or not you would agree with Amar.

1. The discrete random variable \(X\) has probability distribution

where \(a\) and \(b\) are constants.

1. Write down an equation for \(a\) and \(b\).
2. Calculate \(\mathrm { E } ( X )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 3.5\)
3. 1. find a second equation in \(a\) and \(b\),
   2. hence find the value of \(a\) and the value of \(b\).
4. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 5 - 3 X\)
5. Find \(\mathrm { P } ( Y > 0 )\).
6. Find
   1. \(\mathrm { E } ( Y )\),
   2. \(\operatorname { Var } ( Y )\).

3. A discrete random variable \(X\) has the probability function shown in the table below.

1. Given that \(\mathrm { E } ( X ) = \frac { 5 } { 6 }\), find \(a\).
2. Find the exact value of Var ( \(X\) ).
3. Find the exact value of \(\mathrm { P } ( X \leq 15 )\).

2. The random variable \(X\) has probability distribution

1. Given that \(\mathrm { E } ( X ) = 3.5\), write down two equations involving \(p\) and \(q\). Find
2. the value of \(p\) and the value of \(q\),
3. \(\operatorname { Var } ( X )\),
4. \(\operatorname { Var } ( 3 - 2 X )\).

3. A discrete random variable \(X\) has a probability function as shown in the table below, where \(a\) and \(b\) are constants. Given that \(\mathrm { E } ( X ) = 1.7\),

1. find the value of \(a\) and the value of \(b\). Find
2. \(\mathrm { P } ( 0 < X < 1.5 )\),
3. \(\mathrm { E } ( 2 X - 3 )\).
4. Show that \(\operatorname { Var } ( X ) = 1.41\).
5. Evaluate \(\operatorname { Var } ( 2 X - 3 )\).