Chancing your arm is an Hiberno-English and British English term that means 'to take a wild gamble against all odds'. Its origins go back to mediæval Ireland and a famous dispute between two leading noble families, the Butlers (Earls of Ormond) and the FitzGeralds (Earls of Kildare) where the solution came about because one of the leaders quite literally 'chanced his arm'.
The event occured in 1492, when Black James, nephew of the Earl of Ormond, fleeing from FitzGerald's Geraldine soldiers, took sanctuary in the Chapter House of Christchurch Cathedral in Dublin. Though he had the upper hand, with his soldiers surrounding Black James and his men, Gearoid Óg FitzGerald, Ireland's premier earl, wished to end the bloody feud between both families. He pleaded with Black James through the Chapter House's oak door to meet him to negotiate a peace. Black James rebutted all requests. In a historically famous act, FitzGerald ordered his soldiers to cut a hole in the centre of the door. Then, having explained how he wished to see peace between the families, the Earl thrust his hand and arm through the hole to shake hands with Black James. It was a risky venture; any of Black James's heavily armed men could have hacked the Earl's arm off. But FitzGerald's risky gamble against the odds succeeded, in the process producing the term chancing your arm, ironically in a language, english, that none of the participants spoke. (They spoke a mediæval form of the Irish language.)
Though the Chapter House has long been demolished, its restored foundations, at the side of Christchurch Cathedral, are now a tourist attraction. The door of the Chapter House is now on display in St. Patrick's Cathedral a short distance away, where it is now known as the Door of Reconciliation.