This idiom or expression has to do with:

Having an injection, as in:
A. Have you got your shot yet?
B. I’m not going to. I don’t believe in getting flu shots!!

Used with "at," having a chance or an opportunity, as in:
A. Why are you directing such a small budget movie?
B. This is my first shot at directing any kind of movie, and I’m not gonna miss it!!

This idiom or expression means:

Not a good chance. (Is used with a negative verb.)

A. Do you think he has a chance to get admitted to the university at all?
B. He hasn’t got a Chinaman’s chance!!

Also:
A ghost of a chance.
Chinaman’s chance in hell.
A snowball’s chance in hell.

This idiom or expression has to do with:

Taking a risk.

She’s taking a big chance quitting her job in this economy. Either she’s very brave, or she’s simply crazy!!

Also:
Risking it.

This idiom or expression means:

No way.
Not a chance.

A. They want you to apologize.
B. They want ME to apologize? Fat chance!! There’s no way I’ll apologize to them.

Also see:
Chinaman’s chance.

This idiom or expression means:

Some chance.
An opportunity.

Give him half a chance. Take it easy on him.

This idiom or expression like phrase means:

A chance.
An opportunity.

They deserve the opportunity to prove themselves. Give them a fair shot.

This idiom or expression means:

A slight chance.
A chance to win, but with difficulty.

A. Are you gonna try for the team?
B. No, I don’t have a fighting chance.
A. Then I won’t either.
B. No, you should. You can still do it, but you really have to work at it.