Question

If a machine is correctly set up it produces 90% acceptable items. If it is incorrectly set up it produces only 40% acceptable item. Past experience shows that 80% of the setups are correctly done. If after a certain set up, the machine produces 2 acceptable items, find the probability that the machine is correctly set up.

Answer 1

Let A be the event that the machine produces two acceptable items.
Also, let E₁ represent the event that the machine is correctly set up and E₂ represent the event that the machine is incorrectly set up

$$\begin{gathered} \therefore P\left(E_1\right)=0.8 \\ P\left(E_2\right)=0.2 \\ \mathrm{Now}, \\ P\left(A/E_1\right)=0.9\times 0.9=0.81 \\ P\left(A/E_2\right)=0.40\times 0.40=0.16 \\ \mathrm{Using}\mathrm{Bayes}\text{'}\mathrm{theorem},\mathrm{we}\mathrm{get} \\ \mathrm{Required}\mathrm{probability}=P\left(E_1/A\right)=\frac{P\left(E_1\right)P\left(A/E_1\right)}{P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right)} \\ =\frac{{\displaystyle 0.8}\times 0.81}{0.8\times 0.81+0.2\times 0.16} \\ =\frac{81}{85} \end{gathered}$$