In an experiment a group of children each repeatedly throw a dart at a target - Edexcel - A-Level Maths Mechanics - Question 3 - 2018 - Paper 1

To find $P(H > 4)$, we can use the complement rule:

$P(H > 4) = 1 - P(H \leq 4) = 1 - \sum_{k=0}^{4} P(H = k).$
Using the binomial formula:
$P(H = k) = \binom{n}{k} p^k (1-p)^{n-k},$
where $n=10$ and $p=0.1$. Thus, we calculate:

1. $P(H = 0) = \binom{10}{0} (0.1)^0 (0.9)^{10} = 0.3487$
2. $P(H = 1) = \binom{10}{1} (0.1)^{1} (0.9)^{9} = 0.3874$
3. $P(H = 2) = \binom{10}{2} (0.1)^{2} (0.9)^{8} = 0.1937$
4. $P(H = 3) = \binom{10}{3} (0.1)^{3} (0.9)^{7} = 0.0574$
5. $P(H = 4) = \binom{10}{4} (0.1)^{4} (0.9)^{6} = 0.0128$

Summing these probabilities:
$P(H \leq 4) = 0.3487 + 0.3874 + 0.1937 + 0.0574 + 0.0128 = 1.0000$
Then,
$P(H > 4) = 1 - 1.0000 = 0.0128.$

For $P(F = 5)$, using the formula for geometric distribution, we have:
$P(F = n) = (1-p)^{n-1} p,$
where $p = 0.1$ and $n = 5$. Thus:
$P(F = 5) = (0.9)^{4} (0.1) = 0.6561 \times 0.1 = 0.06561.$

To find $\alpha$, we note that the sum of probabilities must equal 1:
$P(F = n) = 0.01 + (n - 1)\alpha.$
Setting up the equation with $n = 2$ to $10$:
$\sum_{n=2}^{10} \left( 0.01 + (n - 1)\alpha \right) = 1.$
Calculating,

Using Thomas' model, we substitute $n = 5$:
$P(F = 5) = 0.01 + (5 - 1) \alpha = 0.01 + 4 \times 0.0253 = 0.01 + 0.1012 = 0.1112.$

Peta’s model assumes that the probability of hitting the target is constant across all throws, applying a binomial distribution. This means each throw is independent with a constant success probability. On the other hand, Thomas' model suggests that the probability increases with each throw, reflecting a scenario where successive attempts yield higher success rates. Hence, while Peta’s model focuses on consistency, Thomas' introduces a variable, dynamic approach to success probability.