"the likelihood that the twenty-first century will not be much like the twentieth" ---> this means that because of what happened in the twentieth century, the writer expects that the 21st century will also have a lot of political change. And therefore it will not be the same the 20th century. If we break it down the writer - I'm guessing historian? says: -They studied past history -There was a lot of political change in the 20th century -They expect that there will be political change in the 21st century. -They are almost 100% sure that it will change -- (the inevitability) --- if something is inevitable, it means that it will happen

"Likelihood" is an expression of probability. The greater the likelihood, the greater the probability. I think you can use the two words interchangeably in most cases. For example: If you eat chocolate chip cookies for every meal the likelihood/probability of weight gain is much greater than if you eat lettuce.

Just want to add that the sentence is unnecessarily wordy. Simply put, Our history suggests (or Based on our history,) there will be political change and it's likely the 21st century will be different from the 20th (or will be unlike the 20th).

I'll add that all history is in the past. Therefore 'past history' sounds ridiculous. A lot of people make this mistake. It simply 'history'

"the likelihood that the twenty-first century will not be much like the twentieth" ---> this means that because of what happened in the twentieth century, the writer expects that the 21st century will also have a lot of political change. If we break it down the writer - I'm guessing historian? says: -They studied past history -There was a lot of political change in the 20th century -They expect that there will be political change in the 21st century. -They are almost 100% sure that it will change -- (the inevitability) --- if something is inevitable, it means that it will happen