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Binomial: green traffic light mornings

Short answer

Using binomial p=0.2: probability exactly one green in 5 mornings=0.4096; exactly four in 20 mornings=0.2182.

A particularly long traffic light on your morning commute is green 20% of the time that you approach it. Assume that each morning represents as independent trial. i) Over five mornings, what is the probability that the light is green on exactly one day? ii) Over 20 mornings, what is the probability that the light is green on exactly four days?

Each morning the light is green with probability p = 0.2 (q = 0.8), independently, so the number of green days X is binomial.

i) n = 5, exactly one green day

P(X = 1) = C(5, 1)(0.2)¹(0.8)⁴ = 5 × 0.2 × 0.4096 = 0.4096

ii) n = 20, exactly four green days

P(X = 4) = C(20, 4)(0.2)⁴(0.8)¹⁶ = 4845 × 0.0016 × 0.028147 = 0.2182

Answer: (i) 0.4096, (ii) 0.2182

✓ Answer checked by Claude Sonnet 5.5

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