- The discrete random variable \(X\) represents the score when a biased spinner is spun. The probability distribution of \(X\) is given by
| \(x\) | - 2 | - 1 | 0 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | \(p\) | \(p\) | \(q\) | \(\frac { 1 } { 4 }\) | \(p\) |
where \(p\) and \(q\) are probabilities.
- Find \(\mathrm { E } ( X )\).
Given that \(\operatorname { Var } ( X ) = 2.5\)
- find the value of \(p\).
- Hence find the value of \(q\).
Amar is invited to play a game with the spinner.
The spinner is spun once and \(X _ { 1 }\) is the score on the spinner.
If \(X _ { 1 } > 0\) Amar wins the game.
If \(X _ { 1 } = 0\) Amar loses the game.
If \(X _ { 1 } < 0\) the spinner is spun again and \(X _ { 2 }\) is the score on this second spin and if \(X _ { 1 } + X _ { 2 } > 0\) Amar wins the game, otherwise Amar loses the game. - Find the probability that Amar wins the game.
Amar does not want to lose the game.
He says that because \(\mathrm { E } ( X ) > 0\) he will play the game. - State, giving a reason, whether or not you would agree with Amar.