# Commute lateness model

## Assumptions

- Let T be commute duration in minutes and b be the departure lead time.
- On a fine day (probability 1-p), T ~ Normal(30, 5^2).
- On an accident day (probability p), T ~ Normal(60, 10^2), representing an occasional roughly doubled commute.
- The sensitivity analysis varies p over 0.02, 0.05, 0.10, and 0.20 because “occasionally” is not a numerical probability.
- A day is late when T > b. Therefore P(late | b,p) = 1 - [(1-p) Phi((b-30)/5) + p Phi((b-60)/10)].
- Required buffer for a target late probability alpha is the numerical quantile b = F_T^{-1}(1-alpha).
- This is a planning model, not an estimate from observed commute logs. Normal components allow negligible negative-time probability and are used only as a transparent approximation; replacing them with truncated/lognormal components would change the tails.

## Interpretation

At b=30 minutes, the fine-day component is centered exactly on the deadline, so about half of fine days are late even before the accident tail is added. The table reports the resulting risk and the buffers required for several standards.
