A-Level Mathematics May/June 2024 Q5(c): In a different area in the Arctic, the probability that a week is a white week is 0.8 .…

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  Let $X$ be the number of white weeks in a sample of 60. Then $X$ follows a binomial distribution, $X \sim B(60, 0.8)$.

  We need to find the probability $P(X < 47)$.

  Since $n$ is large, we can use a Normal approximation. We check the conditions: $np = 60 \times 0.8 = 48$ $nq = 60 \times 0.2 = 12$ Since both $np > 5$ and $nq > 5$, a Normal approximation is suitable.

  The parameters for the Normal approximation $Y$ are: Mean, $\mu = np = 48$ Variance, $\sigma^2 = npq = 60 \times 0.8 \times 0.2 = 9.6$ So, $Y \sim N(48, 9.6)$.

  We want to find $P(X < 47)$, which is equivalent to $P(X \le 46)$. Applying the continuity correction, we find the probability for the continuous variable $Y$: $P(X \le 46) \approx P(Y < 46.5)$

  Now we standardise this value: $Z = \frac{Y - \mu}{\sigma} = \frac{46.5 - 48}{\sqrt{9.6}} = \frac{-1.5}{3.09838...} = -0.4841...$

  We need to find $P(Z < -0.4841)$. Using the symmetry of the Normal distribution: $P(Z < -0.4841) = P(Z > 0.4841)$ $= 1 - P(Z < 0.4841)$ $= 1 - \Phi(0.4841)$

  From the tables, $\Phi(0.4841) = 0.6858$.

  So, the probability is: $1 - 0.6858 = 0.3142$

  Therefore, the probability that fewer than 47 weeks are white is $0.314$ (to 3 s.f.).

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  How the marks are awarded

  • B1 — Correctly stating or using the mean $\mu = 48$ and variance $\sigma^2 = 9.6$ for the Normal approximation.
  • M1 — Applying the continuity correction by converting the discrete probability $P(X < 47)$ to the continuous probability $P(Y < 46.5)$.
  • M1 — Correctly standardising the value, by substituting their continuity-corrected value, mean, and standard deviation into the z-score formula: $z = (46.5 - 48) / \sqrt{9.6}$.
  • M1 — Finding the correct tail probability for their negative z-score, which requires using the rule $P(Z < -z) = 1 - \Phi(z)$. The final probability must be less than 0.5.
  • A1 — Obtaining the correct final answer of 0.314 or a value that rounds to it.

  Common mistakes

  • Forgetting to apply the continuity correction, and instead calculating $P(Y < 47)$, which leads to an incorrect z-score and final answer.
  • Applying the continuity correction incorrectly, for example using $47.5$ instead of $46.5$. Remember that 'fewer than 47' means '46 or less', so the upper boundary is 46.5.
  • Incorrectly finding the probability for a negative z-score. For $z = -0.4841$, a common error is to look up $\Phi(0.4841)$ and give that as the answer, forgetting that $P(Z < -z) = 1 - \Phi(z)$.
  • Using the Poisson approximation, which is not suitable here as $p=0.8$ is not small (the condition is typically $p < 0.1$ and $n > 50$).

  Examiner tip: When approximating a discrete distribution like the Binomial with a continuous one like the Normal, always remember to apply a continuity correction by adding or subtracting 0.5 from the discrete value.

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