2 questions from 2025
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 \( \dfrac{2}{5} \).
If a bulb is chosen randomly from the bulbs produced by \( M_3 \), then the probability that it is defective is ___________.
Three students \( S_1,S_2 \) and \( S_3 \) are given a problem to solve. Consider the following events:
\( U \): At least one of \( S_1,S_2 \) and \( S_3 \) can solve the problem.
\( V \): \( S_1 \) can solve the problem, given that neither \( S_2 \) nor \( S_3 \) can solve the problem,
\( W \): \( S_2 \) can solve the problem and \( S_3 \) cannot solve the problem, \( T \): \( S_3 \) can solve the problem.
For any event \( E \), let \( P(E) \) denote the probability of \( E \). If \( P(U)=\dfrac{1}{2} \), \( P(V)=\dfrac{1}{10} \), and \( P(W)=\dfrac{1}{12} \), then \( P(T) \) is equal to:
