91Ó°ÊÓ

The probability that factory A produces dreamboat cars P(A) = 0.2

The probability that factory B produces dreamboat cars P(B) = 0.5

The probability that factory B produces dreamboat cars P(C) = 0.3

\(\begin{aligned}{}P\left( {lemons\left| A \right.} \right) &= 0.05\\P\left( {lemons\left| B \right.} \right) &= \,0.02\\P\left( {lemons\left| C \right.} \right) &= 0.10\end{aligned}\)

The probability that the lemon dreamboat is produced by factory A, is given by the formula

\(P\left( {A\left| {lemons} \right.} \right) = \frac{{P\left( {lemons\left| A \right.} \right)P\left( A \right)}}{{P\left( {lemons} \right)}}\)

\(\begin{aligned}{}P\left( {lemons} \right) &= P\left( {lemons\left| A \right.} \right)P\left( A \right) + P\left( {lemons\left| B \right.} \right)P\left( B \right) + P\left( {lemons\left| C \right.} \right)P\left( C \right)\\ &= 0.05 \times 0.2 + 0.02 \times 0.5 + 0.10 \times 0.3\\ &= 0.01 + 0.01 + 0.03\\ &= 0.05\end{aligned}\)

\(P\left( {A\left| {lemons} \right.} \right) = \frac{{P\left( {lemons\left| A \right.} \right)}}{{P\left( {lemons} \right)}}\)Therefore,

\(\begin{aligned}{}P\left( {A\left| {lemons} \right.} \right) &= \frac{{P\left( {lemons\left| A \right.} \right)P\left( A \right)}}{{P\left( {lemons} \right)}}\\ &= \frac{{0.05 \times 0.2}}{{0.05}}\\ &= \frac{{0.01}}{{0.05}}\\&= 0.20\end{aligned}\)

Thus the required probability is 0.20.