Three robots

In a workshop, three robots, Q, R, and S, are employed to make chairs    
Robot Q makes 25% of the chairs  
Robot R makes 45% of the chairs  
The remaining chairs are made by Robot S    

Evidence has shown that 2 percent of the chairs made by robot Q are defective, 3 percent of the chairs made by robot R, and 5 percent of the chairs made by robot S are defective.    Construct a tree diagram that illustrates all possible outcomes and probabilities. A chair is randomly selected.    
What is the probability that the chair that robot Q made is defective? 
What is the probability of findings a broken chair?
Given that a chair is defective, what is the probability that it was not made by robot R?

Correct answer:

p₁ =  0.02
p₂ =  0.0335
p₃ =  0.597

Step-by-step explanation:

$p_1=2\%=\frac{2}{100}=\frac{1}{50}=0.02$

$$\begin{gathered} Q=25\%=\frac{25}{100}=\frac{1}{4}=0.25 \\ R=45\%=\frac{45}{100}=\frac{9}{20}=0.45 \\ S=1-Q-R=1-0.25-0.45=\frac{3}{10}=0.3 \\ {p}_2=Q\cdot 0.02+R\cdot 0.03+S\cdot 0.05=0.25\cdot 0.02+0.45\cdot 0.03+0.3\cdot 0.05=0.0335 \end{gathered}$$

$$\begin{gathered} PRD\cdot p_2=P(R\cap D) \\ PRD=R\cdot 0.03/p_2=0.45\cdot 0.03/0.0335=\frac{27}{67}\doteq 0.403 \\ {p}_3=1-PRD=1-0.403=0.597 \end{gathered}$$

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