The best of three quality scores

Question

A sensor takes three independent quality readings, each equally likely to be any value between 0 and 1 (a uniform draw). The readings are independent of one another.

What is the probability that the highest of the three readings is below 0.7?

Accepted Answer

"The highest reading is below 0.7" is the same as "every reading is below 0.7," because if even one reading exceeded 0.7 the maximum would too. So we just need all three to clear the hurdle. For a single uniform reading on 0 to 1, the chance of being below 0.7 is simply P(	ext{one reading} < 0.7) = 0.7. With three independent readings, multiply: P(	ext{max} < 0.7) = (0.7)^3 = 0.343. The general rule for n independent uniforms on 0 to 1 is P(	ext{max} < c) = c^n. Raising a number below 1 to a higher power shrinks it, which captures the idea that the more draws you take, the more likely at l…

The best of three quality scores

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