I decide to answer this question after seeing another question of yours. Before that question, I thought you understood the meaning of the sentence correctly. Now I have a feeling that you probably do not.

Detroit took a gamble on Scott Green a year after the Tigers' third-round draft pick took a gamble of his own.

To understand this question, it's probably important to know that both Detroit and Tigers refer to the same team in Major League Baseball. Also, Tigers' third-round draft pick refers to Scott Green himself. Knowing those, we can understand the sentence easily, like this:

Detroit Tigers took a gamble on Scott Green a year after he took a gamble of his own.

Which means one year earlier, Scott Green took a gamble (his gamble). Then, at the moment that sentence was written, Detroit Tigers had taken another gamble. This gamble was a gamble on Scoot Green. (It appears that Detroit Tigers picked Scott Green in the third round of the draft.)

Now, why "of his own"?

When we say "something of someone's own", it means that that something belongs to that someone.

But why do we say "he took a gamble of his own" instead of just "he took his gamble"?

According to Macmillan Dictionary, own is "used for showing that something belongs to a particular person or thing and not to any other". In other words, saying "he took a gamble of his own" is a kind of emphasis. It will give an impression similar to saying "he took a gamble, and that gamble was his, and no one else's".

I decide to answer this question after seeing another question of yours. Before that question, I thought you understood the meaning of the sentence correctly. Now I have a feeling that you probably do not.

Detroit took a gamble on Scott Green a year after the Tigers' third-round draft pick took a gamble of his own.

To understand this question, it's probably important to know that both Detroit and Tigers refer to the same team in Major League Baseball. Also, Tigers' third-round draft pick refers to Scott Green himself. Knowing those, we can understand the sentence easily, like this:

Detroit Tigers took a gamble on Scott Green a year after he took a gamble of his own.

Which means one year earlier, Scott Green took a gamble (his gamble). Then, at the moment that sentence was written, Detroit Tigers had taken another gamble. This gamble was a gamble on Scoot Green. (It appears that Detroit Tigers picked Scott Green in the third round of the draft.)

Now, why "of his own"?

When we say "something of someone's own", it means that that something belongs to that someone.

But why do we say "he took a gamble of his own" instead of just "he took his gamble"?

According to Macmillan Dictionary, own is "used for showing that something belongs to a particular person or thing and not to any other". In other words, saying "he took a gamble of his own" is a kind of emphasis. It will give an impression similar to saying "he took a gamble, and that gamble was his, and no one else's".