6. In a game two spinners are used. The score on the first spinner is given by the random variable \(A\), which has the following probability distribution:
| \(a\) | 1 | 2 | 3 |
| \(\mathrm { P } ( A = a )\) | \(\frac { 1 } { 3 }\) | \(\frac { 1 } { 3 }\) | \(\frac { 1 } { 3 }\) |
- State the name of this distribution.
- Write down \(\mathrm { E } ( A )\).
The score on the second spinner is given by the random variable \(B\), which has the following probability distribution:
| \(b\) | 1 | 2 | 3 |
| \(\mathrm { P } ( B = b )\) | \(\frac { 1 } { 2 }\) | \(\frac { 1 } { 4 }\) | \(\frac { 1 } { 4 }\) |
- Find \(\mathrm { E } ( B )\).
On each player's turn in the game, both spinners are used and the scores on the two spinners are added together. The total score on the two spinners is given by the random variable \(C\).
- Show that \(\mathrm { P } ( C = 2 ) = \frac { 1 } { 6 }\).
- Find the probability distribution of \(C\).
- Show that \(\mathrm { E } ( C ) = \mathrm { E } ( A ) + \mathrm { E } ( B )\).