2 In a quiz, competitors have to match 5 landmarks to the 5 British counties which the landmarks are in. The random variable \(X\) represents the number of correct matches that a competitor gets, assuming that the competitor guesses randomly. The probability distribution of \(X\) is given in the following table.
| \(r\) | 0 | 1 | 2 | 3 | 4 | 5 |
| \(\mathrm { P } ( X = r )\) | \(\frac { 11 } { 30 }\) | \(\frac { 3 } { 8 }\) | \(\frac { 1 } { 6 }\) | \(\frac { 1 } { 12 }\) | 0 | \(\frac { 1 } { 120 }\) |
- Explain why \(\mathrm { P } ( X = 4 )\) must be 0 .
- Explain how the value \(\frac { 1 } { 120 }\) for \(\mathrm { P } ( X = 5 )\) is calculated.
- Draw a graph to illustrate the distribution.
- Find each of the following.
- \(\mathrm { E } ( X )\)
- \(\operatorname { Var } ( X )\)
- Find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
- There are 12 competitors in the quiz. Assuming that they all guess randomly, find the probability that at least one of them gets all five matches correct.