Question

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

Answer 1

Hint : Probability of any given event is equal to the ratio of the favourable outcomes with the total number of the outcomes. Probability is the state of being probable and the extent to which something is likely to happen in the particular favourable situations. Here, we will find the probability for the machine set up correctly and incorrectly.

Complete step-by-step answer :
Let us suppose that A be the event where the machine produces $2$ acceptable items.
Let Event ${E_1}$ be the event where the machine is set up correctly.
And similarly, let Event ${E_2}$ be the event where the machine is not set up correctly.
From the given data –
If it is incorrectly set up, it produces only $40\%$ acceptable items. Past experience shows that $80\%$ of the setups are correctly done and the machine produces $2$ acceptable items.
$P({E_1}) = 0.8 \\\ P({E_2}) = 1 - 0.8 = 0.2 \;$
If a machine is correctly set up, it produces $90\%$ acceptable items.
$P(A/{E_1}) = 0.9 \times 0.9 = 0.81 \\\ P(A/{E_2}) = 0.4 \times 0.4 = 0.16 \;$
Required conditional probability, $P({E_1}/A) = \dfrac{{P({E_1})P(A/{E_1})}}{{P({E_1})P(A/{E_1}) + P({E_2})P(A/{E_2})}}$
Place values in the above equation –
$P({E_1}/A) = \dfrac{{0.8 \times 0.81}}{{(0.8 \times 0.81) + (0.2 \times 0.16)}}$
Simplify the above equation –
$P({E_1}/A) = \dfrac{{0.648}}{{(0.648) + (0.032)}} \\\ \Rightarrow P({E_1}/A) = \dfrac{{0.648}}{{0.680}} \;$
Numerator and denominator are having three digits after decimal point, so it can be written directly as –
$\Rightarrow P({E_1}/A) = \dfrac{{648}}{{680}}$
Take common factors from both the numerator and the denominator and remove it.
$\Rightarrow P({E_1}/A) = \dfrac{{81}}{{85}}$
Hence, the probability that the machine is correctly set up is $\dfrac{{81}}{{85}}$

Note : Here the probability was converted into decimal points but it can also be solved by keeping it as the fraction. Always follow the general formula for probability and basic simplification properties for the fraction and do it wisely. Always remember that the probability of any event lies between zero and one.