2 Four letters are taken out of their envelopes for signing. Unfortunately they are replaced randomly, one in each envelope.
The probability distribution for the number of letters, \(X\), which are now in the correct envelope is given in the following table.
| \(r\) | 0 | 1 | 2 | 3 | 4 |
| \(\mathrm { P } ( X = r )\) | \(\frac { 3 } { 8 }\) | \(\frac { 1 } { 3 }\) | \(\frac { 1 } { 4 }\) | 0 | \(\frac { 1 } { 24 }\) |
- Explain why the case \(X = 3\) is impossible.
- Explain why \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 24 }\).
- Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).