An underfilled bottle, which machine filled it?

Question

A plant fills bottles on two machines. Machine 1 fills $70\%$ of the bottles and underfills $2\%$ of them. Machine 2 fills the remaining $30\%$ and underfills $5\%$ of them. A random bottle is found to be underfilled.

What is the probability Machine 1 filled it?

Accepted Answer

Imagine 1{,}000 bottles and count the underfilled ones from each machine. - Machine 1: 700 bottles, 2\% underfilled = 14. - Machine 2: 300 bottles, 5\% underfilled = 15. Total underfilled bottles: 14 + 15 = 29. Of these, 14 came from Machine 1: P(	ext{Machine 1} \mid 	ext{underfilled}) = \frac{14}{29} \approx 0.483. Machine 1 handles more than twice the volume of Machine 2 but underfills at a lower rate, so the two end up contributing almost equal numbers of bad bottles. Takeaway. When a high-volume, low-error source meets a low-volume, high-error source, their defect counts can nearly bala…

An underfilled bottle, which machine filled it?

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