# Given chain conditional probabilities what can one calculate?

Lets say I'm given a table of conditional probabilities. For example, lets say that we have probabilities regarding human age. p(3|2) is probability of an individual reaching age 3 given that she is age 2.

Only thing I have is a chain. p(4|3), p(3|2), -- p(n+1 | n).

What probabilities can I calculate given just these conditional probabilities?

• Can we assume that p(2|3)=1?
– Dan
Dec 10, 2015 at 10:37

Using $P(A\cap B)={P(A)}{P(B|A)}$ you can get ratios of the form $\frac{P(3)}{P(2)},\frac{P(4)}{P(3)},$ etc.
Namely: Let $A=2$ and $B=3$, then
• $P(A\cap B)=P(2\cap 3)=P(3)$, since reaching $2$ and $3$ and reaching $3$ are the same events.
• This gives $P(3)={P(2)}{P(3|2)}$, so $\frac{P(3)}{P(2)}=P(3|2)$ and $P(3|2)$ you know.
Other than that, I think you cannot derive anything meaningful. Ofcourse, if you know some chance $P(n)$ explicitly, you can calculate every other $P(m)$, $m\in\mathbb{N}$.
• $\mathsf P(A\cap B) = \mathsf P(A)\cdot\mathsf P(B\mid A)$ Dec 10, 2015 at 11:09