1 In a game at a fair, players choose 4 countries from a list of 10 countries. The names of all 10 countries are then put in a box and the player selects 4 of them at random. The random variable \(X\) represents the number of countries that match those which the player originally chose.
- Show that the probability that a randomly selected player matches all 4 countries is \(\frac { 1 } { 210 }\).
Table 1 shows the probability distribution of \(X\).
\begin{table}[h]
| \(r\) | 0 | 1 | 2 | 3 | 4 |
| \(\mathrm { P } ( X = r )\) | \(\frac { 1 } { 14 }\) | \(\frac { 8 } { 21 }\) | \(\frac { 3 } { 7 }\) | \(\frac { 4 } { 35 }\) | \(\frac { 1 } { 210 }\) |
\captionsetup{labelformat=empty}
\caption{Table 1}
- Find each of the following.
- \(\mathrm { E } ( X )\)
- \(\operatorname { Var } ( X )\)
- A player has to pay \(\pounds 1\) to play the game. The player gets 40 pence back for every country which is matched.
Find the mean and standard deviation of the player's loss per game. - In order to try to attract more customers, the rules will be changed as follows.
The game will still cost \(\pounds 1\) to play. The player will get 25 pence back for every country which is matched, plus an additional bonus of \(\pounds 100\) if all four countries are matched.
Find the player's mean gain or loss per game with these new rules.