TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Bura)/Übungen 2020W/HW02.6

Widget factory

The local widget factory is having a blowout widget sale. Everything must go, old and new. The factory has 450 old widgets, and 1550 new widgets in stock. The problem is that 12% of the old widgets are defective, and 4% of the new ones are defective as well. It is assumed that widgets are selected at random when an order comes in. You are the first customer since the sale was announced.

1. You flip a fair coin once to decide whether to buy old or new widgets. You order two widgets of the same type, chosen based on the outcome of the coin toss. What is the probability that they both will be defective?

2. Given that both widgets turn out to be defective, what is the probability that they were old widgets?

Lösungsvorschlag von Friday

--Friday Sa 30 Jan 2021 14:48:24 CET

Part a)

Since we flip a coin to decide which widget we buy, the probability is equal for both.

$${\frac {1}{2}}*0.12^{2}+{\frac {1}{2}}*0.04^{2}=0.008\quad \Box$$

Part b)

All we have to do is to devide the probability of getting to defective old widget by getting a defective widget at all.

$${\frac {{\frac {1}{2}}*0.12^{2}}{{\frac {1}{2}}*0.12^{2}+{\frac {1}{2}}*0.04^{2}}}={\frac {0.12^{2}}{0.12^{2}+0.04^{2}}}=0.9\quad \Box$$