A factory has three machines X, Y, and Z producing 1000, 2000 and 3000 bolts per day respectively. the machine X produces 1% defective bolts, Y produces 1.5% and Z produces 2% defective bolts. At the end of a day, a bolt is drawn at random and is found to be defective. What is the probability that this defective bolt has been produced by machine X?

Explanation

To solve the problem, we will use Bayes' theorem to find the probability that a defective bolt was produced by machine X. Let's break down the solution step by step.

### Step 1: Define the events
Let:
- \( A_1 \): The event that the defective bolt is produced by machine X.
- \( A_2 \): The event that the defective bolt is produced by machine Y.
- \( A_3 \): The event that the defective bolt is produced by machine Z.
- \( D \): The event that a bolt drawn is defective.
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