Porisma

Worked solution

Carry an underweight-chocolate event into a box count

A factory produces chocolate which are packed into boxes of 20 and delivered to shops for sale. A chocolate will not meet the minimum criteria for packing for sale if it weighs less than 20 grams. On average, 2% of the chocolate produced did not meet the minimum criteria.

(i) Find the probability that a randomly chosen box contain at least 1 chocolate that does not meet the minimum criteria. marks

(ii) Find the probability that out of 4 randomly chosen boxes of chocolate, there are exactly 2 boxes with at least 1 chocolate that does not meet the minimum criteria. marks

[2007/TPJC/II/5]

CJC · Review Problems · Question 8

Answer

(i) 0.3320.332; (ii) 0.2950.295.

(i)

Let X∼B(20,0.02)X\sim B(20,0.02) count underweight chocolates in one box, under the source independent/common-probability model. Then q=P(X≥1)=1−0.9820≈0.3323920282449q=P(X\geq1)=1-0.98^{20}\approx0.3323920282449. The source prints 0.33239202825 and rounds the answer to 0.3320.332 (3 s.f.).

How did this part go?

(ii)

For independent boxes, let Y∼B(4,q)Y\sim B(4,q) count boxes containing at least one underweight chocolate. Then P(Y=2)=(42)q2(1−q)2≈0.2954578119=0.295P(Y=2)=\binom42q^2(1-q)^2\approx0.2954578119=0.295 to 3 significant figures. Carry the unrounded q=1−0.9820q=1-0.98^{20} into this second model.

How did this part go?

Common mistakes

Use the box-level probability in the second model, not the individual-chocolate probability .02.

Read the hint