The work of Harper and subsequent authors has shown that finite sequences $(a_0, \cdots , a_n)$ arising from combinatorial problems are often such that the polynomial $A(z):= \sum_{k=0}^n a_k z^k$ has only real zeros. Basic examples include rows from the arrays of binomial coefficients, Stirling numbers of the first and second kinds, and Eulerian numbers. Assuming the $a_k$ are non-negative, $A(1) > 0$ and that $A(z)$ is not constant, it is known that $A(z)$ has only real zeros iff the normalized sequence $(a_0/A(1), \cdots , a_n/A(1))$ is the probability distribution of the number of successes in $n$ independent trials for some sequence of success probabilities. Such sequences $(a_0, \cdots , a_n)$ are also known to be characterized by total positivity of the infinite matrix $(a_{i-j})$ indexed by non-negative integers $i$ and $j$. This papers reviews inequalities and approximations for such sequences, called {\em P{\'o}lya frequency sequences} which follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates.