In a quiz, a team gains 10 points for every question it answers correctly and loses 5 points for every question it does not answer correctly - Edexcel - A-Level Maths Statistics - Question 5 - 2015 - Paper 1

To score 0 points, the team must answer all questions incorrectly. The probability is given by:

$ext{P(X=0)} = (0.4)^3 = 0.064.$

To score 30 points in 2 rounds, the team has to score 30 points in one of the rounds. Therefore, we need to calculate the probability of scoring 30 points in a single round:

P(X=30) = 0.216.

To get the total for 2 rounds, the probabilities can be computed as: $P(total=30) = P(X=30 in first round) + P(X=30 in second round)$ = 0.216 + 0.216 - P(X=30 in both, the overlap)
Assuming independence, P(X=30 both) = 0.216 * 0.216 = 0.046656

Total probability: 0.216 + 0.216 - 0.046656 = 0.385344

To find E(X), the expected value, we calculate:

$E(X) = ext{sum of each value} \times ext{its probability}$

Specifically:

egin{align*} E(X) & = (30 imes 0.216) + (15 imes 0.432) + (0 imes 0.064) + (-15 imes 0.288)
& = 6.48 + 6.48 + 0 - 4.32
& = 12 \end{align*}

To find Var(X), we will first need E(X^2). Calculate:

$E(X^2) = ext{sum of each value squared} \times ext{its probability}$

egin{align*} E(X^2) & = (30^2 imes 0.216) + (15^2 imes 0.432) + (0^2 imes 0.064) + (-15^2 imes 0.288)
& = (900 imes 0.216) + (225 imes 0.432) + 0 - (225 imes 0.288)
& = 194.4 + 97.2 - 64.8
& = 226.8 \end{align*}

Now, use the formula for variance:

$Var(X) = E(X^2) - (E(X))^2$ = 226.8 - (12)^2 = 226.8 - 144 = 82.8

In the bonus round, the expected value is calculated similarly:

$E(X_{bonus}) = (20 imes P(correct)) + (-5 imes P(incorrect))$ where P(correct) is the probability of getting a question correct.

Probabilities: $P(correct) = 0.6, P(incorrect) = 0.4$

So, $E(X_{bonus}) = 20 imes 0.6 + (-5) imes 0.4 = 12 - 2 = 10$

Thus, the expected points in the bonus round = 10.