For each machine, multiply the number of bolts produced by the defect percentage (as a decimal) to find how many defective bolts that machine produces.

Once you know how many defective bolts each machine made, think: what fraction of all defective bolts came from Machine Z?

In Desmos, type:

• x = 3000*0.012
• y = 5000*0.018
• z = 2000*0.025

and note the values of x, y, and z (the defective counts for X, Y, and Z).

In Desmos, type total = x + y + z to get the total number of defective bolts from all machines.

In Desmos, type p = z / total to get the probability that a randomly chosen defective bolt came from Machine Z. Then, in separate lines, type each answer choice as a fraction (for example, 25/88, 9/44, 5/22, 1/4) and compare their decimal values to p to see which one matches.

We are told the selected bolt is defective and asked for the probability that it came from Machine Z.

This is a conditional probability:

An easier way in this context is to work with counts: among all defective bolts, what fraction were produced by Machine Z?

Use "number produced × percent defective" for each machine.

• Machine X: $3000 \times 1.2\% = 3000 \times 0.012 = 36$ defective bolts.
• Machine Y: $5000 \times 1.8\% = 5000 \times 0.018 = 90$ defective bolts.
• Machine Z: $2000 \times 2.5\% = 2000 \times 0.025 = 50$ defective bolts.

So the defective counts are 36 from X, 90 from Y, and 50 from Z.

Add the defective bolts from all three machines:

So there are 176 defective bolts in total. Among these, 50 are from Machine Z.

The conditional probability is therefore

Now simplify $\dfrac{50}{176}$.

Both 50 and 176 are divisible by 2:

$25$ and $88$ have no common factor other than 1, so $\dfrac{25}{88}$ is fully simplified. This matches choice B, so the correct answer is $\dfrac{25}{88}$.