best braking technique as one approaches red light

I am stumped by the following problem:
I'm traveling at a speed V when I see a red light ahead
at distance d with probability to change to green as function of time p(t), which for example could be gaussian.
I would like to find the velocity profile v(t) that maximizes my velocity (averaged over all scenarios knowing p(t)) as I pass the light, with the condition that I have to stop if I hit the light when its still red.
It doesnt seem to fit the form of a standard functional
cauchy-riemann type problem. Has anyone an insight how to solve it?
.

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