  We address the question of the infinitude of twin and cousin prime pairs from
a probabilistic perspective. Our approach partitions the set of integer numbers
greater than $2$ in finite intervals of the form $[p_{n-1}^2,p_n^2)$, $p_{n-1}$
and $p_n$ being two consecutive primes, and evaluates the probability $q_n$
that such an interval contains a twin prime and a cousin prime. Combining
Merten's third theorem with the properties of the binomial distribution, we
show that $q_n$ approaches $1$ as $n \to \infty$. A study of the convergence
properties of the sequence $\{q_n\}$ allows us to propose a new, more stringent
conjecture concerning the existence of infinitely many twin and cousin primes.
In accord with the Hardy-Littlewood conjecture, it is also shown that twin and
cousin primes share the same asymptotic distribution.
