If there is a 75% probability of accidents in an hour on a highway, what is the chance of accidents in 30 minutes? How many hours does it take for the highway to have an accident almost certain (i.e., 99%)?
We’ll do it in two different ways: first, the analytical.
Let p be the probability of at least one accident in 30 minutes. Then 1 – p is the chance of having no accidents in 30 minutes. Since accidents at a given moment are independent of some other moment, we can apply the AND rule, and the probability of no accidents in one hour is a joint probability of having no accidents in two consecutive half hours.
(1-p) x (1-p) = 1 – 0.75
(1-p)2 = 0.25
1 – p = 0.5
p = 0.5 = 50%
Another way to estimate these probabilities is to apply the Poisson distribution.
1 - ppois(q = 0, lambda = 1.4)0.75where,
q: number of successes
lambda: average rate of success
We adjusted lambda to match 0.75, which turned out to be 1.4 (accidents per hour). To get the probability for 30 minutes, use the same function for the rate of 1.4/2 accidents per half an hour.
1 - ppois(q = 0, lambda = 0.7)0.50341499% chance
Increase the lambda until the 1 – ppois reaches 0.99.
1 - ppois(q = 0, lambda = 4.9)0.9924.9 / 0.7 = 7 times 30 minutes = 3.5 hours.
Or
1- (0.5)7 = 0.99
