Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their production share in the market is $\displaystyle 60 \%, 30 \%$ and $\displaystyle 10 \%$ respectively. Of their respective production capacities, $\displaystyle 20 \%, 10 \%$ and $\displaystyle 5 \%$ cars respectively are electric (or battery operated) . Based on the above, answer the following:

(a)

What is the probability that a randomly selected car is an electric car?

$\displaystyle 37$(i)(a)Let A = Amber manufactures the carB = Bonzi manufactures the carC = Comet manufactures the carE = The selected car is electric$\displaystyle P(A) = \dfrac{60}{100},\; P(B) = \dfrac{30}{100},\; P(C) = \dfrac{10}{100}$$\displaystyle P(E) = P(A) \times P\left(\dfrac{E}{A}\right) + P(B) \times P\left(\dfrac{E}{B}\right) + P(C) \times P\left(\dfrac{E}{C}\right)$$\displaystyle = \dfrac{60}{100} \times \dfrac{20}{100} + \dfrac{30}{100} \times \dfrac{10}{100} + \dfrac{10}{100} \times \dfrac{5}{100}$$\displaystyle = \dfrac{155}{1000}\ or\ \dfrac{31}{200}$$\displaystyle 37$(i)(b)Let A = Amber manufactures the carB = Bonzi manufactures the carC = Comet manufactures the carE = The selected car is a petrol car$\displaystyle P(A) = \dfrac{60}{100},\; P(B) = \dfrac{30}{100},\; P(C) = \dfrac{10}{100}$$\displaystyle P(E) = P(A) \times P\left(\dfrac{E}{A}\right) + P(B) \times P\left(\dfrac{E}{B}\right) + P(C) \times P\left(\dfrac{E}{C}\right)$$\displaystyle = \dfrac{60}{100} \times \dfrac{80}{100} + \dfrac{30}{100} \times \dfrac{90}{100} + \dfrac{10}{100} \times \dfrac{95}{100}$$\displaystyle = \dfrac{845}{1000}\ or\ \dfrac{169}{200}$$\displaystyle 37$(ii)\[P\left(\frac{C}{E}\right) = \frac{P(C) \times P\left(\dfrac{E}{C}\right)}{P(E)} \]\[= \frac{\dfrac{10}{100} \times \dfrac{5}{100}}{\dfrac{60}{100} \times \dfrac{20}{100} + \dfrac{30}{100} \times \dfrac{10}{100} + \dfrac{10}{100} \times \dfrac{5}{100}} \]\[= \frac{\dfrac{50}{10000}}{\dfrac{1550}{10000}} = \frac{1}{31} \]$\displaystyle 37$(iii)\[P\left(\frac{A \text{ or } B}{E}\right) = 1 - P\left(\frac{C}{E}\right) = 1 - \frac{1}{31} = \frac{30}{31} \]