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

To find $P(H > 4)$, we calculate $1 - P(H \leq 4)$ using the binomial formula: $P(H = k) = \binom{n}{k} p^k (1-p)^{n-k},$ where $n=10$ and $p=0.1$. This requires calculating $P(H = 0)$ through $P(H = 4)$ and summing these probabilities, then subtracting from 1.

For Peta's model, the probability of the dart first hitting the target on the 5th throw is given by: $P(F = 5) = P(H = 0) \times P(H = 1) \times P(H = 2) \times P(H = 3) \times P(H = 4).$ These probabilities can be calculated using the binomial distribution mentioned above.

To find $\alpha$, we consider the situation for $n = 10$. Given the condition that $P(F = n)$ must remain valid (i.e., orall n, $P(F = n) \leq 1$), we can substitute $P(F = 10)$ into the equation: $P(F = 10) = 0.01 + (10 - 1) \alpha \leq 1.$ Solving for $\alpha$, we get: $$\alpha \leq \frac{0.99}{9} = 0.11.$

Using Thomas' model: P(F = 5) = 0.01 + (5 - 1) \alpha \quad ext{(from the earlier calculation of $\\alpha$)}. Substituting the found value of $\alpha$, we can evaluate $P(F = 5)$ accordingly.

Peta’s model follows a binomial distribution, implying a consistent probability of hitting the target across all throws, which models the situation realistically for repeated throws. In contrast, Thomas’ model assumes a linear increase in hitting probability with each throw, suggesting that children may improve the more they try, which does not align with the independence assumption of dart throws.