The 438-20 vote in Parliament, which came after the opposition Labour Party dropped its resistance, provided the starting point of an unpredictable six-week race that could change Britain’s relationship to the world.

The election, for Mr. Johnson, is a bet that he and his Conservative Party can win a parliamentary majority. That is uncertain.

Possible outcomes: Though Mr. Johnson’s Conservative Party leads in the polls — more than 10 percentage points in some — it is difficult to predict how that would translate to seats in Britain’s electoral system.

Britain could end up in the same political cul-de-sac it is in today, with no party winning a clear majority, or the Labour leader, Jeremy Corbyn, could ride a wave of public disgust over government deadlock and end up in the prime minister’s office. His party would likely open up a second referendum on the whole departure.

Other potential surprises: Mr. Johnson also has to worry about his right flank, where the more conservative Brexit Party could steal votes with its even more ruthless stance on Britain’s swift departure from the E.U.