To find the probability that the coin shows a head given that Jodie’s score is 8, we use conditional probability:

\(P(H | 8) = \frac{P(H \cap 8)}{P(8)}\)

First, calculate \(P(8)\):

\(P(8) = P(H \text{ and } 4+4) + P(T \text{ and } 2 \times 4) + P(T \text{ and } 4 \times 2)\)

\(= \frac{1}{3} \times \frac{1}{16} + \frac{2}{3} \times \frac{1}{16} + \frac{2}{3} \times \frac{1}{16}\)

\(= \frac{1}{48} + \frac{2}{48} + \frac{2}{48}\)

\(= \frac{5}{48}\)

Next, calculate \(P(H \cap 8)\):

\(P(H \cap 8) = P(H \text{ and } 4+4) = \frac{1}{3} \times \frac{1}{16} = \frac{1}{48}\)

Now, substitute these into the conditional probability formula:

\(P(H | 8) = \frac{\frac{1}{48}}{\frac{5}{48}} = \frac{1}{5}\)