2.04a Discrete probability distributions

3 Two fair dice are thrown. Let the random variable \(X\) be the smaller of the two scores if the scores are different, or the score on one of the dice if the scores are the same.

1. Copy and complete the following table to show the probability distribution of \(X\).

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

2. Find \(\mathrm { E } ( X )\).

7 A vegetable basket contains 12 peppers, of which 3 are red, 4 are green and 5 are yellow. Three peppers are taken, at random and without replacement, from the basket.

1. Find the probability that the three peppers are all different colours.
2. Show that the probability that exactly 2 of the peppers taken are green is \(\frac { 12 } { 55 }\).
3. The number of green peppers taken is denoted by the discrete random variable \(X\). Draw up a probability distribution table for \(X\).

6 Every day Eduardo tries to phone his friend. Every time he phones there is a \(50 \%\) chance that his friend will answer. If his friend answers, Eduardo does not phone again on that day. If his friend does not answer, Eduardo tries again in a few minutes' time. If his friend has not answered after 4 attempts, Eduardo does not try again on that day.

1. Draw a tree diagram to illustrate this situation.
2. Let \(X\) be the number of unanswered phone calls made by Eduardo on a day. Copy and complete the table showing the probability distribution of \(X\).

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

3. Calculate the expected number of unanswered phone calls on a day.

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.

3 A spinner has 5 sides, numbered 1, 2, 3, 4 and 5 . When the spinner is spun, the score is the number of the side on which it lands. The score is denoted by the random variable \(X\), which has the probability distribution shown in the table.

1. Find the value of \(p\). A second spinner has 3 sides, numbered 1, 2 and 3. The score when this spinner is spun is denoted by the random variable \(Y\). It is given that \(\mathrm { P } ( Y = 1 ) = 0.3 , \mathrm { P } ( Y = 2 ) = 0.5\) and \(\mathrm { P } ( Y = 3 ) = 0.2\).
2. Find the probability that, when both spinners are spun together,
   1. the sum of the scores is 4,
   2. the product of the scores is less than 8 .

7 Susan has a bag of sweets containing 7 chocolates and 5 toffees. Ahmad has a bag of sweets containing 3 chocolates, 4 toffees and 2 boiled sweets. A sweet is taken at random from Susan's bag and put in Ahmad's bag. A sweet is then taken at random from Ahmad's bag.

1. Find the probability that the two sweets taken are a toffee from Susan's bag and a boiled sweet from Ahmad's bag.
2. Given that the sweet taken from Ahmad's bag is a chocolate, find the probability that the sweet taken from Susan's bag was also a chocolate.
3. The random variable \(X\) is the number of times a chocolate is taken. State the possible values of \(X\) and draw up a table to show the probability distribution of \(X\).

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.

2 The faces of a biased die are numbered \(1,2,3,4,5\) and 6 . The random variable \(X\) is the score when the die is thrown. The following is the probability distribution table for \(X\). The die is thrown 3 times. Find the probability that the score is 4 on not more than 1 of the 3 throws.

4 A box contains 2 green sweets and 5 blue sweets. Two sweets are taken at random from the box, without replacement. The random variable \(X\) is the number of green sweets taken. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

3 A particular type of bird lays 1,2,3 or 4 eggs in a nest each year. The probability of \(x\) eggs is equal to \(k x\), where \(k\) is a constant.

1. Draw up a probability distribution table, in terms of \(k\), for the number of eggs laid in a year and find the value of \(k\).
2. Find the mean and variance of the number of eggs laid in a year by this type of bird.

3 Two ordinary fair dice are thrown. The resulting score is found as follows.

• If the two dice show different numbers, the score is the smaller of the two numbers.
• If the two dice show equal numbers, the score is 0 .
  1. Draw up the probability distribution table for the score.
  2. Calculate the expected score.

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.

5 Maryam has 7 sweets in a tin; 6 are toffees and 1 is a chocolate. She chooses one sweet at random and takes it out. Her friend adds 3 chocolates to the tin. Then Maryam takes another sweet at random out of the tin.

1. Draw a fully labelled tree diagram to illustrate this situation.
2. Draw up the probability distribution table for the number of toffees taken.
3. Find the mean number of toffees taken.
4. Find the probability that the first sweet taken is a chocolate, given that the second sweet taken is a toffee.

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.

2 A flower shop has 5 yellow roses, 3 red roses and 2 white roses. Martin chooses 3 roses at random. Draw up the probability distribution table for the number of white roses Martin chooses.

6 Pack \(A\) consists of ten cards numbered \(0,0,1,1,1,1,1,3,3,3\). Pack \(B\) consists of six cards numbered \(0,0,2,2,2,2\). One card is chosen at random from each pack. The random variable \(X\) is defined as the sum of the two numbers on the cards.

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 2 } { 15 }\). \includegraphics[max width=\textwidth, alt={}, center]{556a1cc2-47ef-4ef7-a8f6-42850c303531-08_59_1569_497_328}
2. Draw up the probability distribution table for \(X\).
3. Given that \(X = 3\), find the probability that the card chosen from pack \(A\) is a 1 .

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 \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).