Question

A factory has 10 machines which may need adjustment from time to time during the day. Three of these machines are old, each having probability of 1/11 needing adjustment during the day and 7 are new, having the corresponding probability of 1/21. Assuming that no machine needs adjustment twice on the same day, find the probabilities that on a particular day (i) just 2 old and no new machine need adjustment and (ii) just 2 machines that need adjustment are of the same type.

Answer 1

$\text { Let } p_{1}=\text { Probability that an old machine needs adjustment }=\frac{1}{11} \Rightarrow q_{1}=\frac{10}{11} \text {. }$

$\text { and } p_{2}=\text { Probability that a new machine needs adjustment }=\frac{1}{21} \Rightarrow q_{2}=\frac{20}{21} \text {. }$

$\text { Then } P_{1}(x)=\text { Probability that }^{\prime} x^{\prime} \text { old machines need adjustment }$

$=\left(\begin{array}{l}3 \\x\end{array}\right) p_{1}^{x} q_{1}^{3-x}=\left(\begin{array}{l}3 \\x\end{array}\right)\left(\frac{1}{11}\right)^{x}\left(\frac{10}{11}\right)^{3-x} ; x=0,1,2,3$

$\text { and } P_{2}(x)=\text { Probability that ' } x^{\prime} \text { new machines need adjustment }$

$=\left(\begin{array}{l}7 \\x\end{array}\right) p_{2}^{x} q_{2}^{7-x}=\left(\begin{array}{l}7 \\x\end{array}\right)\left(\frac{1}{21}\right)^{x}\left(\frac{20}{21}\right)^{7-x} ; x=0,1,2, \ldots, 7$

$\text { (i) The probability that just two old machines and no new machine need }$

$\text { djustment is given (by the compound probability theorem) by the expression: }$

$P_{1}(2) \cdot P_{2}(0)=\left(\begin{array}{l}3 \\2\end{array}\right)\left(\frac{1}{11}\right)^{2}\left(\frac{10}{11}\right)\left(\frac{20}{21}\right)^{7}=0.016$

$\text { (ii) Similarly, the probability that just } 2 \text { new machines and no old machine need }$

$\text { adjustment is }$

$P_{1}(0) . P_{2}(2)=\left(\frac{10}{11}\right)^{3} \times\left(\begin{array}{l}7 \\2\end{array}\right)\left(\frac{1}{21}\right)^{2}\left(\frac{20}{21}\right)^{5}=0.028$

$\text { The probabilitv that if iust two machines need adjustment, they are of the same }$

$\text { the is the same as the nrohability that 'either just } 2 \text { old and no new or just } 2 \text { new }$$\text { and } n_{0} \text { old machines need adiustment' }$