5. A biased tetrahedral die has faces numbered \(0,1,2\) and 3 . The die is rolled and the number face down on the die, \(X\), is recorded. The probability distribution of \(X\) is
| \(x\) | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | \(\frac { 1 } { 6 }\) | \(\frac { 1 } { 6 }\) | \(\frac { 1 } { 6 }\) | \(\frac { 1 } { 2 }\) |
If \(X = 3\) then the final score is 3
If \(X \neq 3\) then the die is rolled again and the final score is the sum of the two numbers.
The random variable \(T\) is the final score.
- Find \(\mathrm { P } ( T = 2 )\)
- Find \(\mathrm { P } ( T = 3 )\)
- Given that the die is rolled twice, find the probability that the final score is 3