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

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

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

Using the Binomial distribution B(10, 0.1):

$P(H \leq 4) = P(H = 0) + P(H = 1) + P(H = 2) + P(H = 3) + P(H = 4)$

Calculating separately:

$P(H = k) = \binom{10}{k} (0.1)^k (0.9)^{10-k}$

Substituting the values:

• For k = 0: $P(H = 0) = \binom{10}{0}(0.1)^0(0.9)^{10} = 0.3487$
• For k = 1: $P(H = 1) = \binom{10}{1}(0.1)^1(0.9)^{9} = 0.3874$
• For k = 2: $P(H = 2) = \binom{10}{2}(0.1)^2(0.9)^{8} = 0.1937$
• For k = 3: $P(H = 3) = \binom{10}{3}(0.1)^3(0.9)^{7} = 0.0574$
• For k = 4: $P(H = 4) = \binom{10}{4}(0.1)^4(0.9)^{6} = 0.0119$

So,

$P(H \leq 4) = 0.3487 + 0.3874 + 0.1937 + 0.0574 + 0.0119 = 0.9981$

Thus,

$P(H > 4) = 1 - 0.9981 = 0.0019$

Using Peta's model, we need to find the probability of hitting the target for the first time at the fifth throw:

$P(F=5)= P(H<5) \times P(H=5)$

Calculating;

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

Using previous calculations, we find:

$P(F = 5) = (0.3487)(0.3874)(0.1937)(0.0574)(0.0119) = 0.0656$

To find α, we first understand that the total probability must sum to 1:

$0.01 + (1-1)\times \alpha + (2-1)\times \alpha + ... + (10-1)\times \alpha = 1$

Thus, we can write:

$0.01 + 9\alpha = 1$

Rearranging gives:

$9\alpha = 0.99$

Thus,

$\alpha = \frac{0.99}{9} = 0.11$

Using Thomas' model, we have:

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

Substituting the value of α = 0.11:

$P(F=5) = 0.01 + 4\times 0.11$

This simplifies to:

$P(F=5) = 0.01 + 0.44 = 0.45$

Peta's model assumes a fixed probability of hitting the target for each throw, modeling it as a binomial distribution where the outcome of each throw is independent.

In contrast, Thomas' model introduces a linear progression where the probability adjusts based on the number of previous attempts but caps the maximum throws at 10. This suggests a differing perspective; Peta's approach is based purely on consistent probability, while Thomas's takes into account the number of attempts and introduces a parameter (α) to denote the probability of a successful throw on later attempts.