In this paper, the polar coded probabilistic amplitude shaping (PC-PAS) is investigated in a free space optical (FSO) communication system to combat the fading induced by turbulence. The achievable rate of multiple level coding (MLC) and bit-interleaved coded modulation (BICM) schemes with different distributions are studied in turbulence channels, which proves that the non-uniform distribution can achieve larger achievable rates than the uniform distribution in the FSO turbulence channel. And the PC-PAS techniques based on MLC and BICM are both investigated. For MLC-based PC-PAS, the dynamically frozen bits scheme is adopted and the modification to the labeling rule is proposed to label the non-negative constellation points. For the BICM-based PC-PAS, the exchange scheme is proposed to combine the polar codes and PAS technique. The Block error rate (BLER) is evaluated by the Monte Carlo simulation method. From the results, both the MLC-based and the BICM-based PC-PAS can improve the performance compared to the uniform distribution. And the PC-PAS based on MLC outperforms the PC-PAS based on BICM in the same turbulence condition.