CAIE A-Level Mathematics 5.4.2 Binomial and Geometric Distributions

Practise recognising fixed-trial binomial and first-success geometric settings, stating B(n, p) or Geo(p) and calculating exact, cumulative or waiting-time probabilities.

Exam points

choose B(n, p) for a fixed number of independent trials with constant success probability
choose Geo(p) when X counts the trial of the first success and use p(1 − p)^(r−1)
translate fewer than, at least or by a stated trial into the correct sum or complement

[Maximum number: 3]

The mass of grapes sold per day by a large shop can be modelled by a normal distribution with mean 28 kg . On 10 % of days less than 16 kg of grapes are sold.

The mass of grapes sold on any day is independent of the mass sold on any other day.

12 days are chosen at random.

Find the probability that less than 16 kg of grapes are sold on more than 2 of these 12 days.

Let $X\sim\operatorname{Bin}(12,0.1)$ . Then
$P(X<3)=1-\left[\binom{12}{0}(0.1)^0(0.9)^{12}+\binom{12}{1}(0.1)(0.9)^{11}+\binom{12}{2}(0.1)^2(0.9)^{10}\right] =0.111.$

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5.4.2—Geometric distribution question 1