1 A quiz team of 4 students is to be selected from a group of 7 girls and 5 boys. The team is selected at random from the students in the group. The number of girls in the team is denoted by the random variable \(X\).
- Show that \(\mathrm { P } ( X = 4 ) = \frac { 7 } { 99 }\).
Table 1 shows the probability distribution of \(X\).
\begin{table}[h]
| \(r\) | 0 | 1 | 2 | 3 | 4 |
| \(\mathrm { P } ( X = r )\) | \(\frac { 1 } { 99 }\) | \(\frac { 14 } { 99 }\) | \(\frac { 42 } { 99 }\) | \(\frac { 35 } { 99 }\) | \(\frac { 7 } { 99 }\) |
\captionsetup{labelformat=empty}
\caption{Table 1}
- Find each of the following.
- \(\mathrm { E } ( X )\)
- \(\operatorname { Var } ( X )\)
It is decided that the quiz team must have at least 1 girl and at least 1 boy, but the team is still otherwise selected at random. - Explain whether \(\mathrm { E } ( X )\) would be smaller than, equal to or larger than the value which you found in part (b).