Before moving on to nonlinear relationships, it would be helpful to briefly discuss linear probabilities. Consider the example of flipping a coin. When flipping a coin twice, the probability of getting heads is 1/2, or 50 percent on a given flip. If you flip the coin 10 times, the probability of getting heads is 5/10, or 50 percent on a given flip. In this case, the probability of getting heads increases linearly with the number of flips, but the overall probability -- 50 percent -- remains unchanged.
One common kind of nonlinear probability distribution is the Poisson distribution. The Poisson distribution assumes that events are clustered toward small numbers. For example, if you run an insurance company, you will be interested in the distribution of sick claims among your clients. Most people make a number of small claims, but a few make very big claims. The Poisson distribution captures this result.
Another common example of a nonlinear probability distribution is the logistic distribution. The logistic distribution assumes that events are rare up to a certain threshold, and after that threshold, they increase, forming an S-shaped curve. The adoption of certain products follows this distribution. For example, when Google was competing with Yahoo! and Alta Vista early in its history, its user base was indistinguishable from those other search engines. The user base rapidly grew once Google's product superiority became well-known. If you wanted to design a probability model of what search engine consumers used, a logistic distribution might be a suitable choice.
The probit distribution is sometimes used in conjunction with the logistic distribution, but only uses binary variables. For example, imagine rolling dice, but only winning if you role a six. In this case, the results will be clustered into two groups, winning and losing, and the probability of losing will be much greater than winning. If you wanted to know the probability of winning, a probit model might be a good fit.