# probability equation multiplication understanding

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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## 1 Answer

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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Thank you. This is exactly what is happening in the example I am studying, that the population size stays constant. It is a modelling of a router queue, where a new packet may arrive, and a packet may depart in the same step. Or, i guess, no arrival and departure takes place. The state only changes when an arrival or departure happens. So, shall I understand it like this, that the equation is handling the two cases simultaneously, (1) one arrival one departure OR (2) no arrival no departure. That is: (1) OR (2) -> (1) + (2) –  Osman Khalid Dec 23 '12 at 23:44
so if (1) one arrival and one departure means P(Birth)=1, P(Death)=1, this will make only the "arrival" part of equation equal to 1, i.e., (1)(1) + (1-1)(1-1). Now for case (2) no birth, no death, means P(Birth)=0, P(Death)=0, this will make only the "Death" part of equation equal to 1, i.e., (0)(0) + (1-0)(1-0). Thanks for giving me a start! –  Osman Khalid Dec 23 '12 at 23:52
Yes, looks like you've pretty much got it. –  Ilmari Karonen Dec 23 '12 at 23:59