Sentence examples for probability of ruin from inspiring English sources

The law of total probability with A = {red ball on first draw} and Ac = {black ball on first draw} shows that This equation holds for x = 2, 3,…, m − 2. It also holds for x = 1 and m − 1 if one adds the boundary conditions Q 0) = 1 and Q(m) = 0, which say that if Peter has 0 dollars initially, his probability of ruin is 1, while if he has all m dollars, he is certain to win.

We provide several novel results for optimal design of random fields with abc covariances, random walks in finance, and probabilities of ruins related to shocks (e.g. by earthquakes).

A principal problem of insurance risk theory is to find the probability of ultimate ruin.

Rather surprisingly, under these assumptions the probability of ultimate ruin as a function of the initial fortune x is exactly the same as the stationary probability that the waiting time in the single-server queue with Poisson input exceeds x.

Then the probability of ultimate ruin from initial surplus u is defined as ψ ( u ) = P ( T < ∞ | U ( 0 ) = u ).

By performing a Monte Carlo analysis, risk capital required to keep the probability of financial ruin below a threshold value is identified.

If p = 1/2, the probability of Peter's ruin is 0.9 regardless of the values of x and m.

It is quite difficult to determine the probability of Peter's ruin by a direct analysis of all possible cases.

For x = 1 and m = 10, the probability of Peter's ruin is 0.88, only slightly less than before.

An application of the law of total probability to a problem originally posed by Christiaan Huygens is to find the probability of "gambler's ruin".

However, for x = 100 and m = 1,000, Peter's slight advantage on each trial becomes so important that the probability of his ultimate ruin is now less than 0.02.