2.04a Discrete probability distributions

5 A fair three-sided spinner has sides numbered 1, 2, 3. A fair five-sided spinner has sides numbered \(1,1,2,2,3\). Both spinners are spun once. For each spinner, the number on the side on which it lands is noted. The random variable \(X\) is the larger of the two numbers if they are different, and their common value if they are the same.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 7 } { 15 }\). \includegraphics[max width=\textwidth, alt={}, center]{a3b3ebd1-db9e-4552-9abe-bfdeba786d02-08_69_1569_541_328}
2. Draw up the probability distribution table for \(X\).
3. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

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 \(\operatorname { Var } ( X )\).

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 Three fair six-sided dice, each with faces marked \(1,2,3,4,5,6\), are thrown at the same time, repeatedly. For a single throw of the three dice, the score is the sum of the numbers on the top faces.

1. Find the probability that the score is 4 on a single throw of the three dice.
2. Find the probability that a score of 18 is obtained for the first time on the 5th throw of the three dice.

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 \(- 2,1,2,3\). 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 \(\operatorname { Var } ( X )\).

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 \(\operatorname { Var } ( X )\).

2 An ordinary fair die is thrown repeatedly until a 1 or a 6 is obtained.

1. Find the probability that it takes at least 3 throws but no more than 5 throws to obtain a 1 or a 6 .
   On another occasion, this die is thrown 3 times. The random variable \(X\) is the number of times that a 1 or a 6 is obtained.
2. Draw up the probability distribution table for \(X\).
3. Find \(\mathrm { E } ( X )\).

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 \(\operatorname { Var } ( X )\).

5 Anil is taking part in a tournament. In each game in this tournament, players are awarded 2 points for a win, 1 point for a draw and 0 points for a loss. For each of Anil's games, the probabilities that he will win, draw or lose are \(0.5,0.3\) and 0.2 respectively. The results of the games are all independent of each other. The random variable \(X\) is the total number of points that Anil scores in his first 3 games in the tournament.

1. Show that \(\mathrm { P } ( X = 2 ) = 0.114\).
2. Complete the probability distribution table for \(X\).

   \(x\)	0	1	2	3	4	5	6
   \(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)			0.114	0.207	0.285		0.125

3. Find the value of \(\operatorname { Var } ( X )\).

4 The random variable \(X\) takes each of the values \(1,2,3,4\) with probability \(\frac { 1 } { 4 }\). Two independent values of \(X\) are chosen at random. If the two values of \(X\) are the same, the random variable \(Y\) takes that value. 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 probability that \(Y = 2\) given that \(Y\) is even.

2 A bag contains 5 red balls and 3 blue balls. Sadie takes 3 balls at random from the bag, without replacement. The random variable \(X\) represents the number of red balls that she takes.

1. Show that the probability that Sadie takes exactly 1 red ball is \(\frac { 15 } { 56 }\).
2. Draw up the probability distribution table for \(X\).
3. Given that \(\mathrm { E } ( X ) = \frac { 15 } { 8 }\), find \(\operatorname { Var } ( X )\).

5 Eric has three coins. One of the coins is fair. The other two coins are each biased so that the probability of obtaining a head on any throw is \(\frac { 1 } { 4 }\), independently of all other throws. Eric throws all three coins at the same time. Events \(A\) and \(B\) are defined as follows. \(A\) : all three coins show the same result \(B\) : at least one of the biased coins shows a head

1. Show that \(\mathrm { P } ( B ) = \frac { 7 } { 16 }\).
2. Find \(\mathrm { P } ( A \mid B )\).
   The random variable \(X\) is the number of heads obtained when Eric throws the three coins.
3. Draw up the probability distribution table for \(X\).

4 Three fair 4-sided spinners each have sides labelled 1,2,3,4. The spinners are spun at the same time and the number on the side on which each spinner lands is recorded. The random variable \(X\) denotes the highest number recorded.

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 7 } { 64 }\).
2. Complete the probability distribution table for \(X\).

   \(x\)	1	2	3	4
   \(\mathrm { P } ( X = x )\)		\(\frac { 7 } { 64 }\)	\(\frac { 19 } { 64 }\)

   On another occasion, one of the fair 4 -sided spinners is spun repeatedly until a 3 is obtained. The random variable \(Y\) is the number of spins required to obtain a 3 .
3. Find \(\mathrm { P } ( Y = 6 )\).
4. Find \(\mathrm { P } ( Y > 4 )\).

5 A red spinner has four sides labelled \(1,2,3,4\). When the spinner is spun, the score is the number on the side on which it lands. The random variable \(X\) denotes this score. The probability distribution table for \(X\) is given below.

1. Show that \(p = 0.12\).
   A fair blue spinner and a fair green spinner each have four sides labelled 1, 2, 3, 4. All three spinners (red, blue and green) are spun at the same time.
2. Find the probability that the sum of the three scores is 4 or less.
3. Find the probability that the product of the three scores is 4 or less given that \(X\) is odd.

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 A box contains 10 pens of which 3 are new. A random sample of two pens is taken.

1. Show that the probability of getting exactly one new pen in the sample is \(\frac { 7 } { 15 }\).
2. Construct a probability distribution table for the number of new pens in the sample.
3. Calculate the expected number of new pens in the sample.

3 A company produces small boxes of sweets that contain 5 jellies and 3 chocolates. Jemeel chooses 3 sweets at random from a box.

1. Draw up the probability distribution table for the number of jellies that Jemeel chooses.
   The company also produces large boxes of sweets. For any large box, the probability that it contains more jellies than chocolates is 0.64 . 10 large boxes are chosen at random.
2. Find the probability that no more than 7 of these boxes contain more jellies than chocolates.