I will interpret your question in this way: Are there any example of a theorem which proof uses a probability technique to prove the existence of some object where it is proven that the example exists but its explicit construction is impossible?

An example can be the theorem: Every large even number can be expressed as the sum of a prime and the product of at most b primes.

Rényi proved this theorem using a generalisation of ''the large sieve'' method that can be seen as an application of a probabilist method:

Rényi, A. On the large sieve of Yu. V. Linnik. Compositio Mathematica, 8, 68-75 (1956).

Rényi, A. Un nouveau théorème concernant les fonctions indépendantes et ses applications à la théorie des nombres. J. Math, pures et appi. (9), 28, 137-149 (1949).

Rényi, A. Sur un théorème général de probabilité. Ann. Inst. Fourier, 1, 43-52 (1949).

I will interpret your question in this way: Are there any example of a theorem which proof uses a probability technique to prove the existence of some object where it is proven that the example exists but its explicit construction is impossible?

An example can be the theorem:

Every large even number can be expressed as the sum of a prime and the product of at most b primes.

Rényi proved this theorem using a generalisation of ''the large sieve'' method that can be seen as an application of a probabilist method:

Rényi, A. On the large sieve of Yu. V. Linnik. Compositio Mathematica, 8, 68-75 (1956).

Rényi, A. Un nouveau théorème concernant les fonctions indépendantes et ses applications à la théorie des nombres. J. Math, pures et appi. (9), 28, 137-149 (1949).

Rényi, A. Sur un théorème général de probabilité. Ann. Inst. Fourier, 1, 43-52 (1949).

I will interpret your question in this way: Are there any examples of a theorem which proof uses a probability techniques to prove the existence of some object where it is proven that the example exists but its explicit construction is impossible?

An example can be the theorem: Every large even number can be expressed as the sum of a prime and the product of at most b primes.

Rényi proved this theorem using a generalisation of ''the large sieve'' method that can be seen as an application of a probabilist method:

Rényi, A. On the large sieve of Yu. V. Linnik. Compositio Mathematica, 8, 68-75 (1956).

Rényi, A. Un nouveau théorème concernant les fonctions indépendantes et ses applications à la théorie des nombres. J. Math, pures et appi. (9), 28, 137-149 (1949).

Rényi, A. Sur un théorème général de probabilité. Ann. Inst. Fourier, 1, 43-52 (1949).

I will interpret your question in this way: Are there any example of a theorem which proof uses a probability technique to prove the existence of some object where it is proven that the example exists but its explicit construction is impossible?

An example can be the theorem: Every large even number can be expressed as the sum of a prime and the product of at most b primes.

Rényi proved this theorem using a generalisation of ''the large sieve'' method that can be seen as an application of a probabilist method:

Rényi, A. On the large sieve of Yu. V. Linnik. Compositio Mathematica, 8, 68-75 (1956).

Rényi, A. Un nouveau théorème concernant les fonctions indépendantes et ses applications à la théorie des nombres. J. Math, pures et appi. (9), 28, 137-149 (1949).

Rényi, A. Sur un théorème général de probabilité. Ann. Inst. Fourier, 1, 43-52 (1949).