Construct probability distribution from scenario

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

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

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 \(p\).

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 \(X > 3\).

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

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.

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

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

3 A fair cubical die with faces numbered \(1,1,1,2,3,4\) is thrown and the score noted. The area \(A\) of a square of side equal to the score is calculated, so, for example, when the score on the die is 3 , the value of \(A\) is 9 .

1. Draw up a table to show the probability distribution of \(A\).
2. Find \(\mathrm { E } ( A )\) and \(\operatorname { Var } ( A )\).
3. In a spot check of the speeds \(x \mathrm {~km} \mathrm {~h} ^ { - 1 }\) of 30 cars on a motorway, the data were summarised by \(\Sigma ( x - 110 ) = - 47.2\) and \(\Sigma ( x - 110 ) ^ { 2 } = 5460\). Calculate the mean and standard deviation of these speeds.
4. On another day the mean speed of cars on the motorway was found to be \(107.6 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and the standard deviation was \(13.8 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Assuming these speeds follow a normal distribution and that the speed limit is \(110 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), find what proportion of cars exceed the speed limit.

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

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 .

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 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 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.

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 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 .

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

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 .

4 A fair die with faces numbered \(1,2,2,2,3,6\) is thrown. The score, \(X\), is found by squaring the number on the face the die shows and then subtracting 4.

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