Since winning the right to draft forward Blake Griffin with the first pick of the 2009 NBA draft and subsequently adding point guard Chris Paul, the Los Angeles Clippers have transformed from one of the worst to one of the best teams in the NBA. (Nathan Denette/Associated Press)

Gilbert Arenas once famously claimed that he chose to sign with the Washington Wizards over the Los Angeles Clippers as a free agent in 2003 by going against the results of several coin flips. Arenas later admitted the story was a ruse, but there is no denying that a less publicized coin flip at the 2009 NBA Board of Governors’ meeting dramatically changed the course of both franchises.

The Wizards and Clippers both won 19 games in the 2008-09 season and Washington won a coin flip to earn greater odds at winning the Blake Griffin lottery. But the Wizards would lose out on a rapid rebuild, as the combination of numbers that the Clippers received resulted in them landing the No. 1 overall pick in the 2009 draft.

Four years later as the teams prepare to meet Monday night at Verizon Center, Griffin is set to start in the all-star game for the second year in a row and the Clippers have assembled a championship contender around him. The Wizards are tied for the NBA’s worst record and headed toward another lottery appearance.

“Getting Blake Griffin has definitely changed this franchise and this organization around, to say the least,” said Clippers forward Caron Butler, who was a member of the Wizards at the time of the 2009 draft.

Griffin missed the season after the draft with a fractured left patella, and led the Clippers to just 32 wins as a rookie. But he set the foundation for what came next, with his talent, athleticism and high-flying dunks making a no-hope franchise an attractive destination for free agents — and for a generational point guard seeking to force a trade.

When Chris Paul was dealt to the Clippers last season — with some assistance from NBA Commissioner David Stern, who diverted him away from the Los Angeles Lakers in a controversial trade veto — an organization was transformed from mere novelty to serious title threat.

Getting traded to the Clippers was once considered a punishment but Paul embraced the challenge, in part because he didn’t know all that it entailed. In their first 27 years in Los Angeles, the Clippers had a .349 winning percentage (762-1,420), made five postseason appearances and won just one playoff series.

“I think when I got here, I didn’t care about the history, didn’t think much about the history and anything that happened prior to me being here,” Paul said recently. “I’m not being arrogant or anything by that means, but I wasn’t here. I know how I feel and I know how I approach things, so it didn’t matter to me.”

Paul added that he was more confident about joining the Clippers because of “the groundwork they had already. You had Blake. You had [center] DeAndre Jordan. And since I’ve been here, it’s been amazing.”

Paul is currently out with a bruised kneecap, but even with the Clippers (34-15) having lost six of their past eight games, including five of seven without him, they are having the best season in franchise history. Since acquiring Paul, the Clippers have a .649 winning percentage (74-40).

After winning a title with Dallas in 2011, Butler visited the Clippers and was blown away by the presentation from Coach Vinny Del Negro and Neil Olshey, the team’s general manager at the time. They told Butler that the Clippers were serious about becoming a championship organization “and I was a believer. I can kind of read through situations, and I was real positive about that.”

“I think Blake and DeAndre made the Clippers interesting, like they got something over there. CP, Caron and Chauncey [Billups] and all those guys came next and it was like, ‘Okay, they’re serious,’ ” said Jamal Crawford, a former sixth man of the year who signed last summer, along with Grant Hill, Matt Barnes and Ronny Turiaf. “What they did this summer, I think it changed the whole perception of the Clippers. I came here on my visit like, ‘The Clippers got all this?’ ”

Reserve forward Lamar Odom spent his first four seasons with the Clippers before scurrying to Miami the moment he became a free agent, calling the team “basketball hell.” Odom returned to the Clippers in a June trade from Dallas and hardly feels like he has come back to the same organization.

When Odom arrived in 1999, the team practiced at the gym of a small community college. Now it has a spacious, state-of-the-art practice facility, built in Playa Vista prior to the selection of Griffin. And instead of having a roster of young players competing for minutes, points and acclaim, the Clippers now have veterans who have achieved success and are eager to win.

“It’s totally different,” said Odom, who won two championships with the crosstown Lakers. “We got the perfect mixture of a Finals MVP over there [Billups], Grant Hill, players like Chris Paul, players like myself that have played championship ball. Got guys that are locked in and going to be here for a while. First time, you thought that you were going to lose. Now we’re fighting to have the best record in the league.”

The Wizards (11-35) won the draft lottery in 2010 but have continued to struggle in subsequent seasons, going just 54-140 (.278) since selecting John Wall and unable to acquire an all-star player of the caliber of Paul. Butler said he is surprised to see his former team continue to struggle, because “they are very talented” and he actually credits the challenges of his last full season in Washington for some of the success he has experienced in Los Angeles.

“Going through that struggle of being on a 19-win team, you learn to appreciate all of the little things,” Butler said. “Having success in Dallas and being on a team like this here, trying to implement all the little things, like preparation, professionalism, consistency. We’ve got that here, along with a lot of talent.”