To what extent is chance an important factor in the events of "Romeo and Juliet"?
Hugely so. If you closely analyse why the play is a tragedy, it stands alone among Shakespeare's plays. Why do they actually end up dead? Is it because of the Capulet and Montague feud? No. It's because a letter doesn't get delivered, and Romeo arrives at the tomb all of five minutes before his wife wakes up. It's not like Macbeth, who kills the king. It's not like Lear who takes a decision to split up his kingdom. It's not like Othello who listens to Iago.
These two people just fall in love. By chance, at a party. They fall in love. Then, Mercutio is killed - and in fury, Romeo kills Tybalt. Is Tybalt looking for a fight with Mercutio? No. That he happens to be there, and that, in the fumble when Romeo tries to pull the two apart, he receives a mortal wound is chance. It all hinges on chance.
Or, if you like, on fate. The first prologue describes the lovers as "death-mark'd": that the two lovers, before they have even met, are picked out in the stars to end up dead. Of course, that doesn't mean, necessarily that their deaths will be early ones, as many critics sometimes assume. And "death", to the Elizabethans, also meant "orgasm", so death-marked might have had a totally different connotation.
Romeo says something similar in Act 1, Scene 4:
... my mind misgives
Some consequence, yet hanging in the stars,
Shall bitterly begin his fearful date
With this night's revels and expire the term
Of a despised life, clos'd in my breast,
By some vile forfeit of untimely death.
Something hangs in the stars, Romeo thinks, waiting to go wrong, waiting to begin his downward course of action with that same night's party. It will end in the expiration of the life shut up in his breast: with his death. And why? Is it because the Capulet and Montagues don't get on? No. It's because a letter isn't delivered. It's chance. Now that is a modern idea of tragedy. Accidents happen...