How does TOS calculate the POP in its option chain? In particular, I'm interested in the calculation of the probability of a successful trade at expiration for an OTM naked Put. ThinkorSwim shows a value lower than TastyTrade.

TOS and TT likely will state their derivation is proprietary. Google "probability of profit formula" for available information. To "play" with it, you can use Nathan's the free web tool (http://option-risk.appspot.com/OptionRisk.jsp), to get a since of one method, which will provide some insight on the variables and assumptions at play. The "proprietary" element is likely the derivation of the estimate of FRV (Future Realized Volatility) they use. -- Some simply use HV as a proxy for FRV.