CAIE A-Level Mathematics AS 5.5.3 Binomial Distribution Questions

Practise approximating binomial probabilities with a normal distribution and applying continuity correction.

Exam points

identify when a normal approximation to a binomial model is suitable
apply continuity correction when approximating binomial probabilities
calculate and interpret approximate binomial probabilities

[Maximum number: 5]

Salah decides to attempt the crossword puzzle in his newspaper each day. The probability that he will complete the puzzle on any given day is 0.65 , independent of other days.

Use a suitable approximation to find the probability that Salah completes the puzzle more than 50 times in a period of 84 days.

Let $X\sim\operatorname{Bin}(84,0.65)$ . Then
$\mu=84(0.65)=54.6, \qquad \sigma^2=84(0.65)(0.35)=19.11.$
Using the normal approximation and a continuity correction,
$P(X>50)=P\left(Z>\frac{50.5-54.6}{\sqrt{19.11}}\right) =P(Z>-0.9379).$
Therefore,
$P(X>50)=\Phi(0.9379)=0.826.$

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CAIE A-Level Mathematics AS 5.5.3 Binomial Distribution Questions question 1