On Tuesday night, the National Basketball Association (NBA) held their annual draft lottery. In the draft, each team is given the opportunity to select a few players that have declared themselves eligible for the draft (either after completing at least one year of college in the United States or being from another country and over 18 years old). The order of picks in the NBA draft is determined with a goal of awarding earlier picks to teams that performed the worst the previous season. However, rather simply giving the worst team the first pick, second worst the second, etc., the NBA takes the teams that failed to make the playoffs and assigns them a probability of earning the first pick based on their record the previous season. They have been using a probability system based on the previous season’s performance since 1990 and those probabilities can be found here.

The draft lottery was instituted to prevent the worst team from automatically earning the first pick — which allowed the NBA to assuage any fears that a team would intentionally “tank” the end of the season to draft a young stud. From 1985-89, the NBA gave each non-playoff team an equal chance of winning the draft lottery and earning the first pick in the draft. This process began with the New York Knicks earning the first pick in 1985, which allowed the to draft Patrick Ewing, regarded by all the experts as the best available player. Many people accused the NBA of rigging the lottery so the young superstar would be paired with its marquee franchise. But the Knicks never won a championship with Ewing, and he earned a notorious reputation as a guy whose teams improved without him in the lineup (see the Ewing Theory).

From 1990-93, the non-playoff teams were ranked by their records, and the team with the worst record had 11 chances of winning the lottery, the second worst had 10 chances, the third worst had 9 chances, etc, etc, all the way to the eleventh worst team who had one chance. This ended after the 1993 draft lottery, when the Orlando Magic captured the first pick despite having the best record of all the non-playoff teams, and a mere 1.5% chance of winning the lottery. From 1994 to the present, the probability of earning the first pick has been weighted even more toward the teams with the worst records. The worst team has a one in four chance of winning the first pick, and the second worst team has a one in five chance (a full list of the current probabilities can be found here). After the first pick is awarded, the teams with the second and third picks in the draft are also determined using the lottery. The draft order of the remaining teams — those that were not selected in the lottery — are based on their records (the worst teams draft earlier, and the better teams later).

Because the lottery is weighted in favor of the worst teams, sports journalists and commentators are often surprised when the worst teams do not win the first three picks in the draft. Here is how the Associated Press reported it:

Only twice have teams with the worst record won the lottery since the current format began in 1994. Though the lottery is weighted to give teams with the poorest records the best chance to win, the longshots keep finding a way.

That’s 15 draft lotteries, and only two times has the number one pick been awarded to the worst team. Usually, the worst team has a 25% chance of winning the lottery. However, when the Toronto Raptors and Vancouver Grizzlies (now in Memphis) joined the NBA in 1996, the were denied the right to win the first pick until 1999. In 1996 and 1997, the Grizzlies had the worst record in the NBA, so we should really only consider the 13 other seasons. But in 1998, because the Grizzlies and Raptors could not win the lottery, the worst team, the Denver Nuggets, had over a one in three chance of winning (they did not win, and ended up with the third pick in the draft).

If we consider each of the 13 drafts in which the worst team was allowed to win the first pick, we can determine the expected number of draft lotteries won by the worst team. In 12 of those lotteries, the worst team had a 25% chance of winning, and in one draft, they had a 35.92% chance. Therefore, we expect the worst team to earn the first pick in between three and four draft lotteries. What’s the probability they only won two lotteries if the lotteries were fair? It’s about 19.5%, which is comparable to the probability that the worst teams won three and four lotteries (about 25% and 21.5%, respectively). Yes, the worst team has won the lottery less often than expected, but it’s not as an unlikely scenario as the journalists seem to believe.

The other aspect of this is the low probability of the best non-playoff teams winning the lottery. For example, in this year’s lottery, the Chicago Bulls had a 1.7% chance of winning the first pick, yet they’ll be selecting first in the upcoming NBA draft. Other winning teams with low probabilities of success (below 10%) include the Golden State Warriors in 1995 (9.5%), New Jersey Nets in 2000 (4.4%), Houston Rockets in 2002 (8.9%), Milwaukee Bucks in 2005 (6.3%), the Toronto Raptors in 2006 (8.8%), and the Portland Trailblazers in 2007 (5.3%).

What’s the probability that a team with lower than a 10% chance gets the first pick? It’s not 10%. In the current scheme, eight of the 14 lottery teams have less than a 10% chance of winning the lottery. If you combine the probability of winning the first pick for each of those teams, you find that there’s over a 27% chance that one of those teams wins. Therefore, it’s more likely that the number one pick goes to a team with less than a 10% chance of winning than the worst team in the league. There’s also a 12.5% chance that one of the teams with less than a 5% chance of winning the lottery gets the first pick.

How likely is it that a seven teams with less than a 10% chance of winning the lottery got the first pick in the NBA draft? We’ll use this year’s probabilities to save a fair bit of calculation. I invite someone else to try this using the probabilities from each year (I can provide the data). Anyway, when we do the math, we find that there’s less than a 6% chance that the draft lottery will be won seven times by teams with less than a 10% chance of winning in a 15 year span. Here are the probabilities that a team with less than a 10% chance of winning will win in zero through fifteen years of draft lotteries:

As you can see, the most likely outcome is that four teams with less than a 10% chance of winning the draft lottery will get the first pick over 15 years.

Three teams with less than a 5% chance of winning the draft lottery have earned the first pick in the NBA draft. That’s expected in about 18% of all permutations over a 15 year span. The two most likely scenarios are that these teams would win the lottery one or two times. The probabilities of all possible number of times that a team with less than a 5% chance of winning gets the first pick are shown below:

The last two calculations (the ones with the accompanying graphs) should be taken with a grain of salt. As I mentioned, I used only the probabilities from this year. A bit more work — using the probabilities from each of the fifteen years — may reveal that the outcomes of those drafts are not as unlikely as I proposed. However, the take home message here is that, despite what some sportswriters may claim, it’s not all that unlikely to see the worst team (i.e., the one with the highest probability of winning) not win the draft lottery. Additionally, teams that get lucky in the draft lottery, capturing a high draft pick despite a low probability of doing so, are expected very often.

**Related:** A mathematician writes about the probability of the outcome of the 2007 NBA draft lottery.