Porisma

Worked solution

Separate a bulb fault from simultaneous bulb-and-switch faults

When a large number of flashlights leaving a factory is inspected, it is found that the bulb is faulty in 1% of the flashlights and the switch is faulty in 1.5% of them. Assuming that the faults occur independently and at random, find the probability that a sample of

(i) 10 flashlights contains no faulty bulb, marks

(ii) 80 flashlights contains at least one flashlight with both a faulty bulb and a faulty switch. marks

[2009/MJC/II/9(a)]

CJC · Review Problems · Question 2

Answer

(i) 0.9040.904; (ii) 0.01190.0119.

(i)

Let XX count faulty bulbs among 10 flashlights. Then X∼B(10,0.01)X\sim B(10,0.01) and P(X=0)=0.9910≈0.9043820750=0.904P(X=0)=0.99^{10}\approx0.9043820750=0.904 to 3 significant figures.

How did this part go?

(ii)

Independence of the two faults gives P(both)=(0.01)(0.015)=0.00015P(\text{both})=(0.01)(0.015)=0.00015. Let YY count flashlights with both faults among 80, so Y∼B(80,0.00015)Y\sim B(80,0.00015). Then P(Y≥1)=1−P(Y=0)=1−0.9998580≈0.01192917649=0.0119P(Y\geq1)=1-P(Y=0)=1-0.99985^{80}\approx0.01192917649=0.0119 to 3 significant figures.

How did this part go?

Common mistakes

Adding the two fault probabilities would count a union, while part (ii) asks for both faults together.

Read the hint