A cracked egg, which farm did it come from?

Question

A grocer stocks eggs from three farms. Farm 1 supplies $40\%$ of the eggs with a $1\%$ crack rate. Farm 2 supplies $40\%$ with a $2\%$ crack rate. Farm 3 supplies $20\%$ with a $3\%$ crack rate. A random egg is found cracked.

What is the probability it came from Farm 3?

Accepted Answer

Imagine 1{,}000 eggs and count the cracked ones from each farm. - Farm 1: 400 eggs, 1\% cracked = 4. - Farm 2: 400 eggs, 2\% cracked = 8. - Farm 3: 200 eggs, 3\% cracked = 6. Total cracked eggs: 4 + 8 + 6 = 18. Of these, 6 came from Farm 3: P(	ext{Farm 3} \mid 	ext{cracked}) = \frac{6}{18} = \frac{1}{3} \approx 0.333. Farm 3 has the worst crack rate but the smallest supply, so it lands at exactly one third of all cracked eggs. Takeaway. The worst-quality source is not automatically the most likely origin of a defect; its small volume can offset its high rate.

A cracked egg, which farm did it come from?

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