A factory has a total of three manufacturing units, $$M_1$$, $$M_2$$, and $$M_3$$, which produce bulbs independent of each other. The units $$M_1$$, $$M_2$$, and $$M_3$$ produce bulbs in the proportions of $$2:2:1$$, respectively. It is known that $$20\%$$ of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by $$M_1$$, $$15\%$$ are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by $$M_2$$ is $$\frac{2}{5}$$.

If a bulb is chosen randomly from the bulbs produced by $$M_3$$, then the probability that it is defective is ______.

Let the events be defined as follows:
  $$M_1, M_2, M_3$$ : the bulb is produced by unit $$M_1, M_2, M_3$$ respectively.
  $$D$$ : the bulb is defective.

Production proportions (given as $$2:2:1$$) give the prior probabilities
$$P(M_1)=\frac{2}{5},\quad P(M_2)=\frac{2}{5},\quad P(M_3)=\frac{1}{5}$$

Overall defective rate in the factory is $$P(D)=0.20$$.

Defective rates for two of the units are known:
$$P(D\mid M_1)=0.15,\qquad P(M_2\mid D)=\frac{2}{5}=0.40$$

Step 1: Find $$P(D\mid M_2)$$ using Bayes’ theorem.
Bayes’ theorem gives $$P(M_2\mid D)=\frac{P(D\mid M_2)P(M_2)}{P(D)}$$ Therefore $$P(D\mid M_2)=\frac{P(M_2\mid D)\,P(D)}{P(M_2)} =\frac{0.40\times0.20}{\tfrac{2}{5}} =\frac{0.08}{0.40}=0.20$$

Step 2: Use the overall defective rate to solve for $$P(D\mid M_3)$$. The law of total probability gives $$P(D)=P(D\mid M_1)P(M_1)+P(D\mid M_2)P(M_2)+P(D\mid M_3)P(M_3)$$ Substituting the known numbers: $$0.20=0.15\left(\frac{2}{5}\right)+0.20\left(\frac{2}{5}\right)+P(D\mid M_3)\left(\frac{1}{5}\right)$$ Compute the first two terms: $$0.15\left(\frac{2}{5}\right)=0.06,\qquad 0.20\left(\frac{2}{5}\right)=0.08$$ Thus $$0.20=0.06+0.08+\frac{1}{5}P(D\mid M_3)$$ $$0.20-0.14=\frac{1}{5}P(D\mid M_3)$$ $$0.06=\frac{1}{5}P(D\mid M_3)$$ $$P(D\mid M_3)=0.06\times5=0.30$$

Hence, the probability that a bulb produced by $$M_3$$ is defective is $$0.30$$, which lies in the required range $$0.27-0.33$$.