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Worked solution

Condition on the total length before interpreting two short pickles

Peter Piper guarantees every pickle is at least 8 cm long; his bottles hold pickles of at most 10.5 cm. Pickle length is normal with mean 9 cm and standard deviation 0.8 cm.

(i) Find the probability a random pickle must be discarded because(a) it is below the guaranteed length mark or (b) it does not fit the bottle. mark

(ii) He wants to discard no more than 5% of pickles. Comment on whether this is possible. mark

(iii) Find the probability that both of two randomly selected pickles fail the guarantee, given that their total length is below 16 cm. marks

[2009/ACJC/II/11(part of)]

CJC · Review problems · Question 6

Answer

(i)(a)0.106,(b)0.0304; (ii)not possible under the stated model; (iii)0.290 (3 s.f.).

(i) Evaluate the two discard tails

For X∼N(9,0.82)X\sim N(9,0.8^2), P(X<8)=0.106P(X<8)=0.106 and P(X>10.5)=0.0304P(X>10.5)=0.0304, to 3 significant figures.

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(ii) Compare the short-pickle rate with the target

Already 10.6% are too short, exceeding 5%; under this unchanged length distribution and guarantee, total rejects cannot be reduced to 5%. This is the source’s stated reason, even before including oversized pickles.

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(iii) Write the full conditional quotient

Let B={X1<8,X2<8}B=\{X_1<8,X_2<8\} and T={X1+X2<16}T=\{X_1+X_2<16\}. The required probability is P(B∣T)=P(X1<8,X2<8,X1+X2<16)/P(X1+X2<16)P(B\mid T)=P(X_1<8,X_2<8,X_1+X_2<16)/P(X_1+X_2<16).

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(iii) Evaluate the denominator and simplify the numerator

Assuming independent selected lengths, X1+X2∼N(18,1.28)X_1+X_2\sim N(18,1.28) and P(X1+X2<16)=0.0385499P(X_1+X_2<16)=0.0385499. Both lengths below 8 automatically give total below 16, so B∩T=BB\cap T=B. Then P(B∩T)=P(X1<8)P(X2<8)=0.0111619P(B\cap T)=P(X_1<8)P(X_2<8)=0.0111619, using unrounded tails.

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(iii) Divide within the conditioning population

P(B∣T)=0.0111619/0.0385499≈0.28954≈0.290P(B\mid T)=0.0111619/0.0385499\approx0.28954\approx0.290 to 3 significant figures. A total below 16 does not force each individual length below 8.

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Common mistakes

Keep the joint event in the conditional numerator; do not multiply conditional marginal probabilities.

Solution notes

All asks/subparts/marks, both source tails, rejection argument, complete conditional numerator/denominator, sum model and all source decimal stages retained. Independent selected-pickle assumption explicit.

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