We study the first order phase transition of Euler-Heisenberg-AdS black hole

based on free energy landscape. When the Euler-Heisenberg parameter (quantum

electrodynamics parameter) $a<0$, the small (large) Euler-Heisenberg-AdS black

hole can switch to the large (small) black hole due to the change of the

temperature $T$ or Euler-Heisenberg parameter $a$. The local minimal points

correspond to the local stable black holes, and the system prefers black hole

with the lower Gibbs free energy. Furthermore, we research the probability

distribution of the system states by solving the Fokker-Planck equation. A

higher (lower) $T$ corresponds to a larger probability for a large (small)

black hole. A smaller (larger) $a$ corresponds to a larger probability for a

large (small) black hole. The initial the small (large) black hole state can

have the chance to switch to a large (small) black hole state. The coexistent

small and large black hole states can be acquired for some conditions. We also

consider the first passage process for the small-large black hole phase

transition. A higher peak can be acquired for higher (lower) $T$ or smaller

(larger) $a$ with initial small (large) black hole state. For $0\leq a\leq

\frac{32}{7} Q^2 $, the small-large black hole phase transition can be acquired

with a small $a$. A higher (lower) $T$ corresponds to a larger probability for

a large (small) black hole, which is similar to the case of $a<0$. However, the

probability of small (large) black hole will decrease to zero for a large $a$,

which is different from the case of $a<0$. For a small $a$, a higher peak of

the first passage time can be acquired for higher (lower) $T$ or smaller

(larger) $a$ with initial small (large) black hole state.