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I want to understand this probability equation. I'll be grateful, if someone can help.

$P(\text{Birth})P(\text{Death}) + (1-P(\text{Birth}))(1-P(\text{Death}))$

The above is the transition probability of a birth-death process. Can any one explain, why does these multiplications and additions represent.

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Actually, what you've written looks like the probability that no transition occurs on a given time step.

Specifically, assuming that birth and death events are independent, $(1-P(\text{Birth}))(1-P(\text{Death}))$ is the probability that neither a birth nor a death occurs, i.e. that nothing happens at all, while $P(\text{Birth})P(\text{Death})$ is the probability that both a birth and a death occur during the same time step, which obviously also doesn't change the total population size. Since all the other possibilities (a birth and no death, or a death and no birth) do change the population size, the sum of the first two probabilities is the total probability that the population does not change on this time step.

(Of course, that's all ignoring the possibility that more than one birth or death could occur on the same time step, but presumably that's an assumption explicitly or implicitly made in the definition of your process.)

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    Yes, looks like you've pretty much got it.2012-12-23