The best way to combine these sentences is to make them one complex sentence with a subordinate clause.
I think the best way to combine these two sentences is to make them a complex sentence, where one clause is dependent, or subordinate to, the first. I suggest this because they do not seem to really go together on their own. I suggest one of the following.
Pat Sajak made a wry joke about greed when only one letter remained.
In this case, the second clause—the one beginning with when—is the subordinate clause. It makes more sense to me to combine these two sentences with a time-sensitive word, because that is the only way I can see them making sense.
You can also reverse the order, like this.
When Pat Sajak made a wry joke about greed, only one letter remained.
This basically accomplishes the same thing, so it is a matter of preference. In this case, the subordinate cluase is the first part.