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, a team must answer all 3 questions incorrectly. The probability of scoring 0 points can be calculated as follows:

1. Since the probability of answering incorrectly is 0.4, the probability of scoring 0 points is: $P(X=0) = (0.4)^3 = 0.064$

To score a total of 30 points in 2 rounds, a team must score 30 points in one round and 0 points in the other round. We can calculate this as follows:

1. The probability of scoring 30 points (3 correct answers) in one round is: $P(X=30) = (0.6)^3 = 0.216$
2. Therefore, the probability of scoring 0 points in the other round is: $P(X=0) = 0.064$
3. Since these events are independent, the probability of scoring 30 points in one round and 0 in another is: $P(X=30 \text{ in 1st round} \text{ and } X=0 \text{ in 2nd round}) = 0.216 \times 0.064 = 0.013824$

The expected value E(X) can be computed using the formula: $E(X) = \sum (x \cdot P(X=x))$

1. Substituting the probabilities we have: $E(X) = (30 \times 0.216) + (15 \times 0.432) + (0 \times 0.288) + (-15 \times 0.064)$
2. Calculating each term:
   • $30 \times 0.216 = 6.48$
   • $15 \times 0.432 = 6.48$
   • $0 \times 0.288 = 0$
   • $-15 \times 0.064 = -0.96$
3. Summing these values gives: $E(X) = 6.48 + 6.48 + 0 - 0.96 = 12$

The variance Var(X) can be calculated using the formula:

$Var(X) = E(X^2) - (E(X))^2$

1. First, calculate $E(X^2)$: $E(X^2) = (30^2 \times 0.216) + (15^2 \times 0.432) + (0^2 \times 0.288) + ((-15)^2 \times 0.064)$
   • $30^2 \times 0.216 = 194.4$
   • $15^2 \times 0.432 = 97.2$
   • $0^2 \times 0.288 = 0$
   • $(-15)^2 \times 0.064 = 14.4$
2. Adding these: $E(X^2) = 194.4 + 97.2 + 0 + 14.4 = 306$
3. Now, substituting back: $Var(X) = 306 - (12)^2 = 306 - 144 = 162$

In the bonus round, the scoring system changes. The expected score can be determined similarly:

1. Let Y be the number of points scored in the bonus round, where the probabilities are:
   • Correct answers: 20 points
   • Incorrect answers: -5 points
2. With 3 questions, the probability distribution becomes:
   • Correct 3, Incorrect 0: $P(Y=60) = (0.6)^3 = 0.216$
   • Correct 2, Incorrect 1: $P(Y=40) = 3 \times (0.6)^2 \times (0.4) = 3 \times 0.36 \times 0.4 = 0.432$
   • Correct 1, Incorrect 2: $P(Y=20) = 3 \times (0.6) \times (0.4)^2 = 3 \times 0.6 \times 0.16 = 0.288$
   • Correct 0, Incorrect 3: $P(Y=-15) = (0.4)^3 = 0.064$
3. Thus: $E(Y) = (60 \times 0.216) + (40 \times 0.432) + (20 \times 0.288) + (-15 \times 0.064)$ $E(Y) = 12.96 + 17.28 + 5.76 - 0.96 = 34.04$