"Take his gamble" doesn't have quite the same thing as "take a gamble of his own". Perhaps if I use something a bit more concrete for examples, the distinctions will be clearer. First, the simplest phrase, analogous to "take a gamble":
I caught a fish.
The reason that you often hear this is because it has more general application than the phrase you have quoted. That construction ("did something of one's own") is generally used in contrast to someone else doing something comparable. For example:
He caught a fish, and then I caught a fish of my own.
Now, you can also say this:
He caught his fish, and I caught my fish too.
This has more of an implication that we were both expected to catch a fish. The "my fish" in this case is the fish that I was expected to catch. Here's another example:
Michael Jordan will always get his points, but we can still win if we play our best.
This means that Michael Jordan scores a lot of points, and it is expected that he will do so in the upcoming game as well.
So, the meaning of "take his gamble" and "take a gamble of his own" are a bit different: the first means he is taking a specific gamble (one that has been previously explained or implied), and the second means that someone else is taking a gamble and then he is taking a different (probably similar) one.
If you want to depersonalize this, you can say:
If one is a great basketball player, one will get one's points.
Although this is correct English, it's the sort of thing that Queen Elizabeth might say. It would sound stilted or even pretentious coming from an "average Joe".