Sum or difference of two spinners/dice

6 A fair spinner \(A\) has edges numbered \(1,2,3,3\). A fair spinner \(B\) has edges numbered \(- 3 , - 2 , - 1,1\). Each spinner is spun. The number on the edge that the spinner comes to rest on is noted. Let \(X\) be the sum of the numbers for the two spinners.

1. Copy and complete the table showing the possible values of \(X\).

   	Spinner \(A\)
   \cline { 2 - 6 }		1	2	3	3
   Spinner \(B\)	- 2
   - 2			1
   - 1
   1

2. Draw up a table showing the probability distribution of \(X\).
3. Find \(\operatorname { Var } ( X )\).
4. Find the probability that \(X\) is even, given that \(X\) is positive.

6 A fair red spinner has 4 sides, numbered 1,2,3,4. A fair blue spinner has 3 sides, numbered 1,2,3. When a spinner is spun, the score is the number on the side on which it lands. The spinners are spun at the same time. The random variable \(X\) denotes the score on the red spinner minus the score on the blue spinner.

1. Draw up the probability distribution table for \(X\).
2. Find \(\operatorname { Var } ( X )\).
3. Find the probability that \(X\) is equal to 1 , given that \(X\) is non-zero.

2 A fair 6 -sided die has the numbers \(- 1 , - 1,0,0,1,2\) on its faces. A fair 3 -sided spinner has edges numbered \(- 1,0,1\). The die is thrown and the spinner is spun. The number on the uppermost face of the die and the number on the edge on which the spinner comes to rest are noted. The sum of these two numbers is denoted by \(X\).

1. Draw up a table showing the probability distribution of \(X\).
2. Find \(\operatorname { Var } ( X )\).

5 A fair red spinner has four sides, numbered 1, 2, 3, 3. A fair blue spinner has three sides, numbered \(- 1,0,2\). When a spinner is spun, the score is the number on the side on which it lands. The spinners are spun at the same time. The random variable \(X\) denotes the score on the red spinner minus the score on the blue spinner.

1. Draw up the probability distribution table for \(X\).
2. Find \(\operatorname { Var } ( X )\).

Each of four cards has a number printed on it as shown. Two of the cards are chosen at random, without replacement. The random variable \(X\) denotes the sum of the numbers on these two cards.

1. Show that P\((X = 6) = \frac{1}{6}\) and P\((X = 4) = \frac{1}{3}\). [3]
2. Write down all the possible values of \(X\) and find the probability distribution of \(X\). [4]
3. Find E\((X)\) and Var\((X)\). [5]

Two fair six-sided dice are thrown. The random variable \(X\) denotes the difference between the scores on the two dice. The table shows the probability distribution of \(X\).

\(r\)	0	1	2	3	4	5
P(X = r)	\(\frac{1}{6}\)	\(\frac{5}{18}\)	\(\frac{2}{9}\)	\(\frac{1}{6}\)	\(\frac{1}{9}\)	\(\frac{1}{18}\)

1. Draw a vertical line chart to illustrate the probability distribution. [2]
2. Use a probability argument to show that
   1. P(X = 1) = \(\frac{5}{18}\). [2]
   2. P(X = 0) = \(\frac{1}{6}\). [1]
3. Find the mean value of \(X\). [2]

The random variable \(X\) is defined as the difference (always positive or zero) between the scores when 2 ordinary dice are rolled.

1. Copy and complete the probability distribution table for \(X\). [2]

   \(x\)	0	1	2	3	4	5
   P(\(X = x\))

2. Find the expectation and variance of \(X\). [5]