6. The random variable \(A\) represents the score when a spinner is spun. The probability distribution for \(A\) is given in the following table.
| \(a\) | 1 | 4 | 5 | 7 |
| \(\mathrm { P } ( A = a )\) | 0.40 | 0.20 | 0.25 | 0.15 |
- Show that \(\mathrm { E } ( A ) = 3.5\)
- Find \(\operatorname { Var } ( A )\)
The random variable \(B\) represents the score on a 4 -sided die. The probability distribution for \(B\) is given in the following table where \(k\) is a positive integer.
| \(b\) | 1 | 3 | 4 | \(k\) |
| \(\mathrm { P } ( B = b )\) | 0.25 | 0.25 | 0.25 | 0.25 |
- Write down the name of the probability distribution of \(B\).
- Given that \(\mathrm { E } ( B ) = \mathrm { E } ( A )\) state, giving a reason, the value of \(k\).
The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\)
Sam and Tim are playing a game with the spinner and the die.
They each spin the spinner once to obtain their value of \(A\) and each roll the die once to obtain their value of \(B\).
Their value of \(A\) is taken as their value of \(\mu\) and their value of \(B\) is taken as their value of \(\sigma\). The person with the larger value of \(\mathrm { P } ( X > 3.5 )\) is the winner. - Given that Sam obtained values of \(a = 4\) and \(b = 3\) and Tim obtained \(b = 4\) find, giving a reason, the probability that Tim wins.
- Find the largest value of \(\mathrm { P } ( X > 3.5 )\) achievable in this game.
- Find the probability of achieving this value.
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