In a game, balls are thrown to hit a target. The random variable \(X\) is the number of times the target is hit in five attempts. The probability distribution for \(X\) is shown in the following table.
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|
| \(P(X=x)\) | 0.15 | 0.20 | \(k\) | 0.16 | \(2k\) | 0.25 |
(a) Find the value of \(k\). [2]
The player has a chance to win money based on how many times they hit the target.
The gain for the player, in \($\), is shown in the following table, where a negative gain means that the player loses money.
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|
| Player’s gain (\$) | \(-4\) | \(-3\) | \(-1\) | 0 | 1 | 4 |
(b) Determine whether this game is fair. Justify your answer. [3]
