Reverend Bayes meets Darwin: Bayesian inference of phylogeny and its impact on evolutionary biology

The probability of a high roll is larger for the biased dice than for the fair dice. Suppose that you draw a die at random from the box and roll it twice, observing a four on the first roll and a six on the second roll. What is the probability that the die is biased? A Bayesian analysis combines ones prior beliefs about the probability of a hypothesis with the likelihood. The likelihood is the vehicle that carries the information about the hypothesis contained in the observations. In this case, the likelihood is simply the probability of observing a four and a six given that the die is biased or fair. Assuming independence of the tosses, the probability of observing a four and a six is