Question at position 1 You begin by analyzing the output of climate model simulations with low vs. high carbon emission scenarios. You find an average of 33 major floods per century (i.e., 100 years) under the high carbon emission scenario vs. an average of 21 major floods under the low carbon emission scenario. Compute the probability of at least 7 major floods occurring under the low carbon emission scenario over a given 32-year period. Assume that a maximum of one major flood occurs each year. Answer You begin by analyzing the output of climate model simulations with low vs. high carbon emission scenarios. You find an average of 33 major floods per century (i.e., 100 years) under the high carbon emission scenario vs. an average of 21 major floods under the low carbon emission scenario. Compute the probability of at least 7 major floods occurring under the low carbon emission scenario over a given 32-year period. Assume that a maximum of one major flood occurs each year. [input]

Question

Question at position 1  You begin by analyzing the output of climate model simulations with low vs. high carbon emission scenarios. You find an average of 33 major floods per century (i.e., 100 years) under the high carbon emission scenario vs. an average of 21 major floods under the low carbon emission scenario. Compute the probability of at least 7 major floods occurring under the low carbon emission scenario over a given 32-year period. Assume that a maximum of one major flood occurs each year. Answer You begin by analyzing the output of climate model simulations with low vs. high carbon emission scenarios. You find an average of 33 major floods per century (i.e., 100 years) under the high carbon emission scenario vs. an average of 21 major floods under the low carbon emission scenario. Compute the probability of at least 7 major floods occurring under the low carbon emission scenario over a given 32-year period. Assume that a maximum of one major flood occurs each year. [input]

BlackTom AI · Accepted Answer

The question asks for the probability of at least 7 major floods in a 32-year period under the low carbon emission scenario, given an average of 21 floods per 100 years.

First, convert the long-run average into a yearly rate: 21 floods per 100 years means a rate of 0.21 floods per year. Over 32 years, the expected number of floods is λ = 0.21 × 32 = 6.72.

Assuming floods follow a Poisson process (independence year to year and a maximum of one flood per year aligns with a low-rate Poisson model over the 32-year window), X ~ Poisson(λ = 6.72) represents the number of floods in 32 years. We want P(X ≥ 7) = 1 − P(X ≤ 6).

Now compute the cumulative probability up to 6. The Poisson probability mass function is P(X = k) = e^(−λ) λ^k / k!. Using λ = 6.72:
- P(0) = e^(−6.72) ≈ 0.00121
- P(1) = P(0) × 6.72 ≈ 0.00813
- P(2) = P(1) × 6.72 / 2 ≈ 0.0273
- P(3) = P(2) × 6.72 / 3 ≈ 0.0612
- P(4) = P(3) × 6.72 / 4 ≈ 0.1029
- P(5) = P(4) × 6.72 / 5 ≈ 0.1385
- P(6) = P(5) × 6.72 / 6 ≈ 0.1553
Adding these: P(X ≤ 6) ≈ 0.00121 + 0.00813 + 0.0273 + 0.0612 + 0.1029 + 0.1385 + 0.1553 ≈ 0.495.

Therefore, P(X ≥ 7) = 1 − P(X ≤ 6) ≈ 1 − 0.495 ≈ 0.505. So the probability is about 0.50 (roughly 50%).

Notes and caveats:
- The calculation relies on a Poisson model for rare events over a fixed window; if floods are not well-modeled as Poisson (e.g., clustering or seasonality), the result could differ.
- If one interpreted the yearly cap of one flood as a hard constraint rather than a Poisson process, a different distribution would be needed, but under the standard assumption used here, the result above is the appropriate estimate.

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