A holiday light string has 27 lights, and a whole string fails if and only if on

Question

A holiday light string has 27 lights, and a whole string fails if and only if one or more of its own individual lights fails. Each single light has an independent 2.7% chance of failure during any one year period. You buy two such light strings (one with all red lights, the other all green), and will simultaneously power each string for a full year. At year end, what is the percentage probability that the red string will be bright, and the green string will have failed) [Answer is a single value.]

Explanation / Answer

P(failure of single light) = 0.027

P (no failure for one year ) = 1 - 0.027 = 0.973

and probability of failure of complete holiday light string = 1- P(out of 27 lights, no single light will fail)

P(out of 27 lights, no single light will fail) = (0.973)27 = 0.4776

probability of failure of complete holiday light string = 1- P(out of 27 lights, no single light will fail)

= 1 - 0.4776 = 0.5224

So probability that red string will be bright or will not fail = 0.4776

probability that green string will fail = 0.5224

so probability of these two simultanous independent events getting happened = 0.4776 * 0.5224 = 0.2495 = 0.25 (approx)

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