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 can use the cumulative distribution function (CDF) for a binomial random variable. We calculate:

$P(H > 4) = 1 - P(H \leq 4)$

Calculating $P(H \leq 4)$:

$P(H \leq 4) = \sum_{k=0}^{4} P(H = k) = \sum_{k=0}^{4} {10 \choose k} (0.1)^k (0.9)^{10-k}$

Performing this calculation gives the required answer.

To find $P(F = 5)$, we need to consider that the dart has not hit the target in the first 4 throws and then hits on the 5th:

$P(F = 5) = P(H = 0) \times P(H = 1) \times P(H = 2) \times P(H = 3) \times P(H = 4) \times P(H = 5)$

Using the binomial formula for $P(H = k)$, we can compute this sum.

We know that total probabilities should sum to 1. Hence,

$P(F = 1) + P(F = 2) + \\ldots + P(F = 10) = 1$

Substituting our equations for $P(F=n)$ and solving for $$ gives the value of $\\alpha$.

Using Thomas’ model:

$P(F = 5) = 0.01 + (5 - 1) \times \alpha$

We substitute the previously found value of $\alpha$ into this equation to find $P(F = 5)$.

Peta’s model assumes a binomial distribution based on a fixed probability of hitting the target across independent throws. In contrast, Thomas’ model assumes a more general progression that incorporates a different probability structure, suggesting a non-constant probability of hitting across multiple trials. This means that Thomas allows for the possibility of hitting the target earlier or later in a way that adapts based on the attempts made.