Maybe it's an UK vs US thing whether you can use the "that" construct or not?
"Probability of something happening" or "probability that something will happen" are both correct.
"Probability that something happens" is incorrect, by the strict standards.
Are you sure? How about
"the probability that it rains on Thursdays is 1/7"?
Sounds better to me than
"the probability that it will rain on Thursdays is 1/7"? (i.e. speaking of thursdays in general, not a particular one)
You're right; I actually included that in my explanation but I hadn't paid enough attention to Harvey's statement of what was incorrect. It looks weird, especially in an abstracted, non-real-world context, but it is right in all the ways that matter.
"The probability that something happens will depend upon someone's level of motivation to do it."
It's just hard to use it with a present tense. Your example about raining on Thursdays is close, but by using the present tense, it doesn't quite make sense, because probability is not the same as a statistic; that is to say, whether or not it rained on any given Thursday in the past is not subject to any probability at all, because it either happened or didn't, there can't be a chance that something happened in the past. And of course since English is English, using the present in this case encompasses the past as well.
So the problem there is the word "probability," not "that."
The second sentence is also weird not because of "that" but because of what it means, which is that a vague selection of thursdays in the future will undoubtedly be subject to a certain chance of precipitation. It's a very strange thing to say, and one which could not be known as absolutely as it's presented.
[/linguisticsstudent]