The camera flags a blue jersey

Question

At a packed stadium, 90% of fans wear red and 10% wear blue. A color-recognition camera identifies a jersey's color correctly 85% of the time, meaning for a truly blue jersey it says "blue" 85% of the time, and for a truly red jersey it says "red" 85% of the time (calling it "blue" the other 15%).

The camera flags one fan's jersey as blue.

What is the probability that jersey is actually blue?

Accepted Answer

Let B mean "truly blue." Imagine 1000 fans. - Blue jerseys: 100; the camera calls "blue" on 85\% \Rightarrow 85 true hits. - Red jerseys: 900; the camera mistakenly calls "blue" on 15\% \Rightarrow 135 false alarms. The camera says "blue" for 85 + 135 = 220 fans, of whom only 85 are truly blue: P(B \mid 	ext{"blue"}) = \frac{85}{220} = \frac{17}{44} \approx 0.386. Despite the camera being "85% accurate," a "blue" flag is more likely wrong than right, the huge red majority generates more false alarms than there are true blues. Takeaway. When one category dominates, even a small error rate on …

The camera flags a blue jersey

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