6 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.
- Show that the probability that a randomly selected competitor guesses all 4 correctly is \(\frac { 1 } { 35 }\).
Let \(X\) represent the number of correct guesses made by a randomly selected competitor. The probability distribution of \(X\) is shown in the table.
| \(r\) | 0 | 1 | 2 | 3 | 4 |
| \(\mathrm { P } ( X = r )\) | 0 | \(\frac { 4 } { 35 }\) | \(\frac { 18 } { 35 }\) | \(\frac { 12 } { 35 }\) | \(\frac { 1 } { 35 }\) |
- Find the expectation and variance of \(X\).