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Worked solution

A binomial tail and the least sample size

After the Monetary Authority of Singapore’s report on the structured financial products sold by financial institutions in Singapore, a survey found that 95% of the respondents would favour greater consumer protection.

(i) Find the probability that out of 30 respondents, 25 of them would favour greater consumer protection.

(ii) Find the least number of respondents surveyed such that the probability that at least 40 of them would favour greater consumer protection exceeds 0.980.

[2009/CJC/II/9]

CJC · Discussion · Question 3

Answer

(i) 0.01240.0124. (ii) Least n=46n=46.

(i)

Let X count respondents favouring greater protection among 30. Then X∼B(30,0.95)X\sim B(30,0.95). Thus P(X=25)=(3025)(0.95)25(0.05)5=0.0123530245…=0.0124P(X=25)=\binom{30}{25}(0.95)^{25}(0.05)^5=0.0123530245\ldots=0.0124 (3 s.f.).

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(ii)

Let Y∼B(n,0.95)Y\sim B(n,0.95). Require P(Y≥40)=1−P(Y≤39)>0.980P(Y\geq40)=1-P(Y\leq39)>0.980. The printed table is retained below.

n1−P(Y≤39)
450.97614
460.99249
470.99787

The least n is 46. For fixed p, adding another Bernoulli trial cannot reduce the probability of at least 40 successes; therefore the failure at 45 and success at 46 establish minimality.

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Common mistakes

The threshold is strictly greater than 0.980; check the preceding sample size as well.

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