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 use the complementary probability:

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

We can calculate $P(X \leq 5)$ by summing from 0 to 5:

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

Instead, we can also compute:

$P(X > 5) = 1 - \sum_{k=0}^{5} P(X = k)$

Using the binomial formula:

$P(X > 5) = 1 - \sum_{k=0}^{5} \binom{15}{k} (0.48)^k (0.52)^{15-k}$

After calculating, we find:

$P(X > 5) \approx 0.920\ (or 0.9203)$

To approximate the probabilities for large sample sizes, we can use the normal approximation to the binomial distribution:

Let:

• $Y$ = number of hits, where $Y \sim N(np, np(1-p))$
• $n = 250$
• $p = 0.48$

Calculating the mean and standard deviation:

• Mean:
  $np = 250 \times 0.48 = 120$

• Variance:
  $np(1-p) = 250 \times 0.48 \times 0.52 \approx 30$

• Standard Deviation:
  $\sigma = \sqrt{30} \approx 5.477$

Now convert the target of more than 110 hits to a Z-score:

$Z = \frac{X - \mu}{\sigma} = \frac{110 - 120}{5.477} \approx -1.826$

Using the standard normal distribution table to find the probability of Z being greater than -1.826, we can find:

$P(Y > 110) = 1 - P(Z \leq -1.826) \approx 0.8854$