NEW YORK — It may be a case of too little, too late. Even with consecutive wins over the Yankees in the first two games in the Bronx, the Red Sox still trail their divisional rivals by 5 1/2 games in the wild card race with eight games left on their slate.

Even so, with Friday’s 10-8 victory and Saturday’s 7-3 triumph, the Sox have now ensured that they will not have to suffer the indignity of watching the Yankees celebrate clinching a playoff berth in their own ballpark. Yet even that does not reflect the full measure of Boston’s ambitions.

After all, as Sox infielder Jed Lowrie said on Friday night when asked if the team was still holding out hope for a postseason berth, “Why not?”

The Sox have four games remaining against New York, along with four in Chicago against the White Sox. The Red Sox do not control their destiny — they would have to border on perfect and hope that New York faltered against the Blue Jays. The odds of reaching the postseason are terrible — coolstandings.com suggests that the Sox have a 0.8 percent shot at overthrowing the Yankees for the wild card — but insofar as the Sox are one of just five teams (the Rangers and Twins, who have both clinched, as well as the Rays and Yankees comprise the rest of the list) not to have been eliminated from a spot in the playoffs in the American League, the Sox are intent on competing.

“We’ve got a chance. That being said, it’s a long road,” said Sox starter Jon Lester. “Weirder things have happened in baseball. We’ve got four more games against the Yankees. We’ll see. All we can do is go out there and play hard and put up a good effort. You’re not going to expect anything less from these guys in this clubhouse. We’re going to go out there and battle and grind it out, and we’ll see where we are in a couple games.”