Final answer:
The success probability in logistic regression with one predictor is modeled by a logistic curve, with the success (p) and failure (q) probabilities summing up to one, as applied in binomial distribution scenarios. The Bayesian approach incorporates Bayes' theorem to provide updated probability estimates based on the data collected.
Step-by-step explanation:
In logistic regression with one predictor, the success probability is modeled by a logistic curve. This curve is defined such that there are two outcomes - success or failure. The probability of success, denoted by p, and the probability of failure, denoted by q, together sum up to one (p + q = 1). When considering a binomial distribution, each trial has a fixed probability p of success, and there is a definite number of trials. For example, if you're playing a game with a 55% chance to win (p = 0.55), and a 45% chance to lose (q = 0.45), for a series of 20 games, the logistic regression model can predict the probability for a number of wins (successes).
Using the Bayesian approach to logistic regression, we could calculate Bayes' theorem to update the probability estimates as more data becomes available. Bayes' theorem is expressed as P(0|x), where 0 represents the parameter estimates, and x the data collected. This allows for the calculation of the updated probability of a model or hypothesis given the data that has been observed.