Peter C. answered • 1d
Commercial Insurance, ERM & Cyber Risk Expert 20+ Years Exp.rie
Binomial Distribution — Farmer Mendez
Summary of the Data
Historical estimate of killing-frost probability: p = 3/14 ≈ 0.214286
Period considered: n = 5 years (a 5-year period)
Farmer wants to be 90% certain that a killing frost will occur no more than 1 year in 5 (i.e., P(X ≤ 1) ≥ 0.90).
This is a classic issue of looking at or having to limited a set of data for the assessment or having to simply go with the data that is reliable.
Binomial Probability Calculations (n=5, p=3/14)
| k (number of frost years in 5 years) | P(X = k) | Rounded (3 d.p.) |
| 0 | 0.299449 | 0.299 |
| 1 | 0.408340 | 0.408 |
| 2 | 0.222731 | 0.223 |
| 3 | 0.060745 | 0.061 |
| 4 | 0.008283 | 0.008 |
| 5 | 0.000452 | 0.000 |
Key Probabilities
P(X = 0) = 0.299449 ≈ 0.299
P(X = 1) = 0.408340 ≈ 0.408
P(X ≤ 1) = P(X=0) + P(X=1) = 0.707789 ≈ 0.708
P(X ≥ 2) = 1 − P(X ≤ 1) = 0.292211 ≈ 0.292
Expected number of frost years in 5 years: E[X] = n·p = 1.0714 ≈ 1.071
Interpretation & Risk-Management Recommendations
Conclusion:
- Based on historical data (p = 3/14), the probability that a killing frost will occur no more than 1 year in a 5-year period is approximately 0.708 (≈ 70.8%). If the failure of the business or the need to purchase insurance or self insure is based on this data then the answer is that the results fall outside of the 1 in 5 yr parameter.
This is well below the farmer's target confidence level of 90%. In other words, given the historical rate, the farmer would only be about 70.5% certain that at most one of the next five years will have a killing frost; there is roughly a 29.5% chance of 2 or more frost years in a 5-year span.
Risk implications:
The farmer faces non-trivial risk of multiple frost years within any 5-year window (≈ 29.5%).
The expected number of frost years over 5 years is about 1.07, so on average the farmer should expect roughly one killing frost every five years, slightly above the farmer's threshold.
Recommended risk-management actions:
Financial Tools:
Investigate crop/freeze insurance products available in your jurisdiction (multi-year or annual policies).
Build a frost contingency reserve (self insure): given the expected frequency, target a reserve sized to cover one year of worst-case damage or the deductible for insurance.
Explore government or NGO disaster assistance programs and eligibility requirements.
Agronomic & Operational Measures:
Shift to frost-hardy or later-maturing varieties when practical.
Adjust planting dates to reduce the chance of sensitive stages coinciding with frost risk windows.
Use protective measures where feasible: frost cloths/row covers, irrigation for frost protection, wind machines/heaters in high-value blocks, or temporary greenhouse/hoop houses for critical crops.
Diversification & Contracting:
Diversify crop mix to reduce exposure of total acreage to frost-sensitive crops in the same year.
Consider contracting a portion of production or using futures/options (where available) to reduce revenue volatility.
Monitoring & Early Warning:
Subscribe to reliable meteorological forecasts and frost alerts; create an operational response plan for frost warnings.
Re-evaluate probability estimates regularly:
Use updated historical data and consider climate trends; if frost frequency is changing, re-calculate p and the binomial probabilities.
Numeric Summary (rounded)
P(X = 0): 0.299
P(X = 1): 0.408
P(X ≤ 1): 0.708 (70.8%)
P(X ≥ 2): 0.292 (29.2%)
Expected frost years in 5 years: 1.071
