To chance your arm is an idiom which means to take a risk in order to gain something that you want. To 'chance your arm at something' can also mean to try something or to give something a go even if there is a risk of failing.

1. When I overheard Lucy saying that she had a spare ticket to the concert, I chanced my arm and asked if I could have it. Luckily, she said yes.
2. I didn't think I would be any good at cricket but I chanced my arm and gave it a go.

1. Even though the company are not advertising any vacant positions, you could always chance your arm and drop your CV in anyway.
2. I know Paul is a risk taker, but I'm worried that he is chancing his arm a bit too much by giving up his well paid job to start his own business.