5.02b Expectation and variance: discrete random variables

3 The possible values of the random variable \(X\) are the 8 integers in the set \(\{ - 2 , - 1,0,1,2,3,4,5 \}\). The probability of \(X\) being 0 is \(\frac { 1 } { 10 }\). The probabilities for all the other values of \(X\) are equal. Calculate

1. \(\mathrm { P } ( X < 2 )\),
2. the variance of \(X\),
3. the value of \(a\) for which \(\mathrm { P } ( - a \leqslant X \leqslant 2 a ) = \frac { 17 } { 35 }\).

7 Judy and Steve play a game using five cards numbered 3, 4, 5, 8, 9. Judy chooses a card at random, looks at the number on it and replaces the card. Then Steve chooses a card at random, looks at the number on it and replaces the card. If their two numbers are equal the score is 0 . Otherwise, the smaller number is subtracted from the larger number to give the score.

1. Show that the probability that the score is 6 is 0.08 .
2. Draw up a probability distribution table for the score.
3. Calculate the mean score. If the score is 0 they play again. If the score is 4 or more Judy wins. Otherwise Steve wins. They continue playing until one of the players wins.
4. Find the probability that Judy wins with the second choice of cards.
5. Find an expression for the probability that Judy wins with the \(n\)th choice of cards.

2 The random variable \(X\) has the probability distribution shown in the table. Two independent values of \(X\) are chosen at random. The random variable \(Y\) takes the value 0 if the two values of \(X\) are the same. Otherwise the value of \(Y\) is the larger value of \(X\) minus the smaller value of \(X\).

1. Draw up the probability distribution table for \(Y\).
2. Find the expected value of \(Y\).

4 The six faces of a fair die are numbered \(1,1,1,2,3,3\). The score for a throw of the die, denoted by the random variable \(W\), is the number on the top face after the die has landed.

1. Find the mean and standard deviation of \(W\).
2. The die is thrown twice and the random variable \(X\) is the sum of the two scores. Draw up a probability distribution table for \(X\).
3. The die is thrown \(n\) times. The random variable \(Y\) is the number of times that the score is 3 . Given that \(\mathrm { E } ( Y ) = 8\), find \(\operatorname { Var } ( Y )\).

4 A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable \(X\) represents the number of paperback books she chooses.

1. Show that the probability that she chooses exactly 2 paperback books is \(\frac { 3 } { 14 }\).
2. Draw up the probability distribution table for \(X\).
3. You are given that \(\mathrm { E } ( X ) = 3\). Find \(\operatorname { Var } ( X )\).

4 Coin \(A\) is weighted so that the probability of throwing a head is \(\frac { 2 } { 3 }\). Coin \(B\) is weighted so that the probability of throwing a head is \(\frac { 1 } { 4 }\). Coin \(A\) is thrown twice and coin \(B\) is thrown once.

1. Show that the probability of obtaining exactly 1 head and 2 tails is \(\frac { 13 } { 36 }\).
2. Draw up the probability distribution table for the number of heads obtained.
3. Find the expectation of the number of heads obtained.

3 In a probability distribution the random variable \(X\) takes the value \(x\) with probability \(k x ^ { 2 }\), where \(k\) is a constant and \(x\) takes values \(- 2 , - 1,2,4\) only.

1. Show that \(\mathrm { P } ( X = - 2 )\) has the same value as \(\mathrm { P } ( X = 2 )\).
2. Draw up the probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
3. Find \(\mathrm { E } ( X )\).

3 Andy has 4 red socks and 8 black socks in his drawer. He takes 2 socks at random from his drawer.

1. Find the probability that the socks taken are of different colours.
   The random variable \(X\) is the number of red socks taken.
2. Draw up the probability distribution table for \(X\).
3. Find \(\mathrm { E } ( X )\).

4 Mrs Rupal chooses 3 animals at random from 5 dogs and 2 cats. The random variable \(X\) is the number of cats chosen.

1. Draw up the probability distribution table for \(X\).
2. You are given that \(\mathrm { E } ( X ) = \frac { 6 } { 7 }\). Find the value of \(\operatorname { Var } ( X )\).

5 A game is played with 3 coins, \(A , B\) and \(C\). Coins \(A\) and \(B\) are biased so that the probability of obtaining a head is 0.4 for coin \(A\) and 0.75 for coin \(B\). Coin \(C\) is not biased. The 3 coins are thrown once.

1. Draw up the probability distribution table for the number of heads obtained.
2. Hence calculate the mean and variance of the number of heads obtained.

6 At a funfair, Amy pays \(\\) 1$ for two attempts to make a bell ring by shooting at it with a water pistol.

• If she makes the bell ring on her first attempt, she receives \(\\) 3\( and stops playing. This means that overall she has gained \)\\( 2\).
• If she makes the bell ring on her second attempt, she receives \(\\) 1.50\( and stops playing. This means that overall she has gained \)\\( 0.50\).
• If she does not make the bell ring in the two attempts, she has lost her original \(\\) 1$.

The probability that Amy makes the bell ring on any attempt is 0.2 , independently of other attempts.

1. Show that the probability that Amy loses her original \(\\) 1$ is 0.64 .
2. Complete the probability distribution table for the amount that Amy gains.

   Amy's gain (\$)
   Probability	0.64

3. Calculate Amy's expected gain.

6 A fair five-sided spinner has sides numbered 1, 1, 1, 2, 3. A fair three-sided spinner has sides numbered \(1,2,3\). Both spinners are spun once and the score is the product of the numbers on the sides the spinners land on.

1. Draw up the probability distribution table for the score. \includegraphics[max width=\textwidth, alt={}, center]{da4a61b9-f55d-40ed-a721-a6aee962f0d6-08_67_1569_484_328}
2. Find the mean and the variance of the score.
3. Find the probability that the score is greater than the mean score.

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

1. Draw up the probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

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 \(b\).

8 A discrete random variable \(X\) has the following probability distribution.

1. Find the value of the constant \(c\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).

6 A box contains five balls numbered \(1,2,3,4,5\). Three balls are drawn randomly at the same time from the box.

1. By listing all possible outcomes (123, 124, etc.), find the probability that the sum of the three numbers drawn is an odd number. The random variable \(L\) denotes the largest of the three numbers drawn.
2. Find the probability that \(L\) is 4 .
3. Draw up a table to show the probability distribution of \(L\).
4. Calculate the expectation and variance of \(L\).

6 In a competition, people pay \(\\) 1\( to throw a ball at a target. If they hit the target on the first throw they receive \)\\( 5\). If they hit it on the second or third throw they receive \(\\) 3\(, and if they hit it on the fourth or fifth throw they receive \)\\( 1\). People stop throwing after the first hit, or after 5 throws if no hit is made. Mario has a constant probability of \(\frac { 1 } { 5 }\) of hitting the target on any throw, independently of the results of other throws.

1. Mario misses with his first and second throws and hits the target with his third throw. State how much profit he has made.
2. Show that the probability that Mario's profit is \(\\) 0$ is 0.184 , correct to 3 significant figures.
3. Draw up a probability distribution table for Mario's profit.
4. Calculate his expected profit.

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

1. Find the value of \(q\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

2 The random variable \(X\) takes the values \(- 2,0\) and 4 only. It is given that \(\mathrm { P } ( X = - 2 ) = 2 p , \mathrm { P } ( X = 0 ) = p\) and \(\mathrm { P } ( X = 4 ) = 3 p\).

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

7 A fair die has one face numbered 1, one face numbered 3, two faces numbered 5 and two faces numbered 6 .

1. Find the probability of obtaining at least 7 odd numbers in 8 throws of the die. The die is thrown twice. Let \(X\) be the sum of the two scores. The following table shows the possible values of \(X\). \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Second throw}

   		1	3	5	5	6	6
   \cline { 2 - 8 }	1	2	4	6	6	7	7
   	3	4	6	8	8	9	9
   First	5	6	8	10	10	11	11
   throw	5	6	8	10	10	11	11
   	6	7	9	11	11	12	12
   	6	7	9	11	11	12	12

   \end{table}
2. Draw up a table showing the probability distribution of \(X\).
3. Calculate \(\mathrm { E } ( X )\).
4. Find the probability that \(X\) is greater than \(\mathrm { E } ( X )\).

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 \(X\).

2 Two unbiased tetrahedral dice each have four faces numbered \(1,2,3\) and 4. The two dice are thrown together and the sum of the numbers on the faces on which they land is noted. Find the expected number of occasions on which this sum is 7 or more when the dice are thrown together 200 times.

5 In a particular discrete probability distribution the random variable \(X\) takes the value \(\frac { 120 } { r }\) with probability \(\frac { r } { 45 }\), where \(r\) takes all integer values from 1 to 9 inclusive.

1. Show that \(\mathrm { P } ( X = 40 ) = \frac { 1 } { 15 }\).
2. Construct the probability distribution table for \(X\).
3. Which is the modal value of \(X\) ?
4. Find the probability that \(X\) lies between 18 and 100 .

7 Sanket plays a game using a biased die which is twice as likely to land on an even number as on an odd number. The probabilities for the three even numbers are all equal and the probabilities for the three odd numbers are all equal.

1. Find the probability of throwing an odd number with this die. Sanket throws the die once and calculates his score by the following method.
   • If the number thrown is 3 or less he multiplies the number thrown by 3 and adds 1 .
   • If the number thrown is more than 3 he multiplies the number thrown by 2 and subtracts 4 .
   The random variable \(X\) is Sanket's score.
2. Show that \(\mathrm { P } ( X = 8 ) = \frac { 2 } { 9 }\). The table shows the probability distribution of \(X\).

   \(x\)	4	6	7	8	10
   \(\mathrm { P } ( X = x )\)	\(\frac { 3 } { 9 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 9 }\)

3. Given that \(\mathrm { E } ( X ) = \frac { 58 } { 9 }\), find \(\operatorname { Var } ( X )\). Sanket throws the die twice.
4. Find the probability that the total of the scores on the two throws is 16 .
5. Given that the total of the scores on the two throws is 16 , find the probability that the score on the first throw was 6 .

2 The discrete random variable \(X\) has the following probability distribution. Given that \(\mathrm { E } ( X ) = 2.3\) and \(\operatorname { Var } ( X ) = 3.01\), find the values of \(p , q\) and \(r\).