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Probabilities of substitution given the Jukes-Cantor model is...

$$P_{ij} = \begin{cases} \frac{1}{4} + \frac{3}{4}\exp({-\mu t}), & \text{for all }i=j \\ \frac{1}{4}[1-\exp({-\mu t})], & \text{for all }i\neq{j} \end{cases}$$

$$P_{ij} = \begin{cases} \frac{1}{4} + \frac{3}{4}\exp({-\mu t}), & \text{for all }i=j \\ \frac{1}{4}-\frac{1}{4}\exp({-\mu t}), & \text{for all }i\neq{j} \end{cases}$$

(Testing best ways to write)

Substitution models and maximum likelihood
Substitution models are central to calculating the likelihood of a phylogenetic tree. A fundamental step in calculating the likelihood is the use of a substitution model to calculate the probability of a finding some state j after a period of time t given that the ancestor is state i. For example, if we are examining DNA we might wish to calculate the probability of finding an A after time t given that the ancestor is also an A; this could reflect the absence of any change or it could reflect a substitution from A to some other nucleotide followed by reversion to an A (or, in principle, any sequence of substitutions that ends as an A). Using the Jukes-Cantor model (often abbreviated as the JC69 model or the JC model) this probability is:

$$P_{ij} = \begin{cases} \frac{1}{4} + \frac{3}{4}\exp({-\mu t}), & \text{for all }i=j \\ \frac{1}{4}-\frac{1}{4}\exp({-\mu t}), & \text{for all }i\neq{j} \end{cases}$$

The Jukes-Cantor model is the simplest model of DNA sequence evolution; it assumes that the frequency of each nucleotide is $$\frac{1}{4}$$ and the instantaneous rates of change among all nucleotides is identical. Time is typically measured in substitutions per site, meaning that the mutation rate $$\mu$$ is 1. Since the Jukes-Cantor model is time symmetrical it is not limited to ancestor-descendent relationships.