George throws a ball at a target 15 times - Edexcel - A-Level Maths Statistics - Question 1 - 2022 - Paper 1

To find P(X > 5), we can use the complement rule:

$P(X > 5) = 1 - P(X \\leq 5)$

We first need to calculate P(X ≤ 5). This can be computed by summing the probabilities from P(X = 0) to P(X = 5) using the binomial formula:

P(X = k) = {15 rack k} (0.48)^k (0.52)^{15 - k}

After computing separately for each value, we find:

$P(X \\leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)$

Using cumulative calculations: $P(X > 5) = 1 - P(X \\leq 5) \\ = 0.920$

For large n, we can use the normal approximation: Let Y be the number of hits which follows: $Y \\sim N(np, np(1 - p))$ Where:

• $n = 250$, $p = 0.48$,
• Mean ($\\mu$) = $np = 250 imes 0.48 = 120$,
• Variance ($\\sigma^2$) = $np(1 - p) = 250 imes 0.48 imes 0.52 = 30.$

Calculating the standard deviation: $\\sigma = \sqrt{30} \\approx 5.48$

To find P(Y > 110): Convert to the z-score: $z = \frac{X - \\mu}{\sigma} = \frac{110 - 120}{5.48} \\approx -1.82$

Using the standard normal distribution, we find: $P(Z > -1.82) \\approx 0.885$

Thus, the approximation gives us: $P(Y > 110) \\approx 0.885$