2 A football player is practising taking penalties. On each attempt the player has a \(70 \%\) chance of scoring a goal. The random variable \(X\) represents the number of attempts that it takes for the player to score a goal.
- Determine \(\mathrm { P } ( X = 4 )\).
- Find each of the following.
- \(\mathrm { E } ( X )\)
- \(\operatorname { Var } ( X )\)
- Determine the probability that the player needs exactly 4 attempts to score 2 goals.
- The player has \(n\) attempts to score a goal.
- Determine the least value of \(n\) for which the probability that the player first scores a goal on the \(n\)th attempt is less than 0.001 .
- Determine the least value of \(n\) for which the probability that the player scores at least one goal in \(n\) attempts is at least 0.999.