(a) Probability of scoring 7:

To score 7, Gustav must get heads on the coin and roll a 6 on the die:

\[ P(7) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12} = 0.0833 \]

(b) Completing the probability table:

Each number from 1 to 6 can be obtained in two ways:

• Final score \(x\) is obtained from rolling \(x\) with tails.
• Final score \(x\) is obtained from rolling \(x-1\) with heads.

Thus, for \( x = 1 \) to \( 6 \):

\[ P(x) = \frac{1}{12} + \frac{1}{12} = \frac{2}{12} = \frac{1}{6} \]

The completed table:

(c) Expected Value Calculation:

Using the expected value formula:

\[ E(X) = (1 \times \frac{1}{12}) + (2 \times \frac{1}{6}) + (3 \times \frac{1}{6}) + (4 \times \frac{1}{6}) + (5 \times \frac{1}{6}) + (6 \times \frac{1}{6}) + (7 \times \frac{1}{12}) \]

Calculating:

\[ E(X) = \frac{1}{12} + \frac{2}{6} + \frac{3}{6} + \frac{4}{6} + \frac{5}{6} + \frac{6}{6} + \frac{7}{12} \]

\[ E(X) = \frac{1}{12} + \frac{4}{12} + \frac{6}{12} + \frac{8}{12} + \frac{10}{12} + \frac{12}{12} + \frac{7}{12} \]

\[ E(X) = \frac{48}{12} = 4 \]

Conclusion: The expected value of Gustav’s final number of points is 4.

…………………………..Markscheme…………………………..

• (a) Correct probability calculation for scoring 7.
• (b) Proper completion of the probability table.
• (c) Correct use of the expected value formula.