In an experiment a group of children each repeatedly throw a dart at a target - Edexcel - A-Level Maths Statistics - Question 3 - 2018 - Paper 2

Using the complement rule: $P(H > 4) = 1 - P(H \\leq 4)$
Calculating P(H ≤ 4) using the binomial distribution: $P(H = k) = \binom{n}{k} p^k (1-p)^{n-k}$ we sum for k = 0, 1, 2, 3, 4: $P(H \\leq 4) = \sum_{k=0}^{4} \binom{10}{k} (0.1)^k (0.9)^{10-k}$
After calculations, we find: $P(H > 4) = 1 - 0.9872 = 0.0128.$

Given P(F = n) = 0.01 × (1 - 0.1)^{n-1} for the first hit, We substitute n = 5: $P(F = 5) = 0.01 \cdot (0.9)^{4} = 0.01 \cdot 0.6561 = 0.006561.$ Thus, P(F = 5) is approximately 0.656.

To find α, we can sum the probabilities for n = 1 to 10: $\sum_{n=1}^{10} P(F = n) = 1.$
The equation becomes: $0.01 + 9\alpha = 1 \\Rightarrow 9\alpha = 0.99 \\Rightarrow \alpha = 0.11.$

Using Thomas' model: $P(F = 5) = 0.01 + (5 - 1) \cdot 0.11 = 0.01 + 0.44 = 0.45.$ Thus, P(F = 5) using Thomas' model is 0.45.

Peta's model assumes that the probability of hitting the target is constant for each throw, while Thomas' model assumes that the probability may increase with each attempt. Essentially, Peta treats each throw independently, whereas Thomas incorporates a learning aspect, suggesting that children improve with repeated attempts.