The Real Jackpot Payoffs for Powerball on November 28th

Powerball is advertising that the expected jackpot for tonight’s drawing is $500 million and that there is a 1 in 175 million chance of matching all six numbers and winning a share of the jackpot.  The ads are misleading because $500 million equals the undiscounted value of payouts received over a 30 year period, not the cash value, and assumes that the prize will not be shared.  A more accurate description of the jackpot prize distribution is:

  • A one in 476 million chance of winning $163.7 million (the jackpot is shared by two winners)
  • A one in 479 million chance of winning $327.4 million (a single jackpot winner)
  • A one in 947 million chance of winning $109.1 million (the jackpot is shared by three winners)
  • A one in 2.82 trillion chance of winning  $81.9 million (the jackpot is shared by four winners)
  • A one in 11.23 trillion chance of sharing the jackpot among at least five winners

If the advertised jackpot increases to $600 million, a more accurate description of the jackpot prize distribution would be:

  • A one in 539 million chance of winning $196.4 million (the jackpot is shared by two winners)
  • A one in 682 million chance of winning $131.0 million (the jackpot is shared by three winners)
  • A one in 852 million chance of winning $392.9 million (a single jackpot winner)
  • A one in 1.29 trillion chance of winning  $98.2 million (the jackpot is shared by four winners)
  • A one in 3.27 trillion chance of sharing the jackpot among at least five winners

The most likely outcomes are that there will be a single winning ticket or the jackpot will be shared by two winners.  As the advertised jackpot nears $600 million, it becomes much more likely that there will be at least a three-way split of the jackpot.  Either way, the best strategy for lottery players is to choose a combination of six numbers that the other 200+ million lottery players won’t select.

Is a Powerball Ticket a Good Bet?

The advertised Powerball jackpot has now been revised up to $500 million and the game rules indicate that the chance of winning the jackpot is 1 in 175 million for each $2 ticket that is purchased.  To the naïve Powerball customer this may seem like the expected payout from a $2 ticket is well above $2.  That reasoning is incorrect because the $500 million payout over 30 years has a cash value of only $327 million and there is a 63% chance that a winning ticket will be shared with one or more other players.  If Powerball lottery officials are correct and the jackpot is $500 million on Wednesday night, the expected payout on a $2 ticket will be $1.53.  If lottery officials have underestimated Wednesday’s jackpot the expected value of a $2 ticket will be somewhat lower.  For example, if the jackpot is $550 million (paid out over 30 years) the expected value of a ticket declines to $1.50.

Lottery officials expect about 227 million $2 tickets to be sold.  32.5% of the proceeds from these ticket sales are added to the cash value of the jackpot with another 17.5% funding consolation prizes.  This suggests there should be a 50% return on a $2 ticket, or a $1 expected value based only on current sales.  The expected return on a $2 ticket is substantially higher than this, about $1.53, because of the players in the 15 previous drawings where no winning ticket was selected.  These prior Powerball players have subsidized players in this week’s drawing.

Although 227 million tickets are expected to be purchased between Saturday night’s drawing and Wednesday’s drawing, with the odds of a winning ticket being 1 in 175 million, there is a  37% chance that there will be no winning ticket on Wednesday night.  If even more than 227 million tickets are sold, the jackpot increases but so does the chance of sharing the prize with other winners.  The following table illustrates some possible outcomes if the jackpot continues to increase for the November 28th Powerball drawing.

Jackpot Values and Expected Returns for November 28 Powerball Drawing
Advertised Jackpot Cash Value of Jackpot Tickets Sold Chance of No Winning Tickets Expected Value of $2 Ticket
$500 million $327 million 176.3 million 36.6% $1.53
$550 million $360 million 226.7 million 27.4% $1.50
$600 million $393 million 277.0 million 20.6% $1.48

A $2 Powerball ticket typically has an expected return of $1 meaning that half of the money spent on tickets will not be paid out in prizes.  Because there have been 15 weeks of drawings with no winners the rollover in the Powerball jackpot is about $203 million.  This means a $2 ticket purchased this week has an expected value of $1.53.  It is more likely than not that ticket sales will be above the projected amount of 227 million, which will raise the jackpot but dilute the expected value of each ticket sold.  The expected value of a ticket will likely remain near $1.50, but as the jackpot rises it becomes much more likely that there will be at least one winner on Wednesday night.  A winner on Wednesday will reduce the advertised jackpot on Saturday December 1 to $40 million (cash value of $25.8 million).

Brees, Unitas, Yaz and Cabrera: Comparing Great Performances Across Generations

Last week Miguel Cabrera earned baseball’s Triple Crown, the first one in 45 years, by leading the American League in home runs, runs batted in and batting average.  Within days of that feat Drew Brees broke an NFL record held by Johnny Unitas for 52 years ago.  Brees has thrown for a touchdown pass in 48 straight games.  Brees is an amazing quarterback, but Cabrera’s accomplishment is more impressive.

Football may be the most popular sport in America, but it is impossible to compare individual accomplishments across eras.  Professional football in 2012 does not resemble the game played from 1956-1960 when Unitas completed at least one touchdown pass in 47 straight games.  NFL teams throw 55% more passes per game than they did in 1956 when Unitas began his streak.  Through the first five games of the 2012 season Brees has thrown 236 passes.  This is more than Bart Starr, Bobby Layne, John Brodie, Billy Wade, Eddie LeBaron, Zeke Bratkowski and three other starting quarterbacks attempted through the entire 1960 season.

The short pass has largely replaced the running play in short yardage situations as teams get close to the goal line.   During Brees’ streak 47% of his touchdown passes have been within 10 yards of the end zone compared to 37% for Unitas during his streak.  In addition, 29% of Brees’ touchdown passes have been from the 5 yard line or closer, compared to 22% for Unitas.

Cabrera’s feat displayed an amazing ability to hit for both power and average in an era of more and more specialization in sports.  One of the biggest changes that hitters confront today is that they must face fresh and talented set-up relievers and closers on a daily basis.  In 1967, when Carl Yastrzemski last won the Triple Crown, the typical team had 39 complete games from their starting pitchers and teams generally used a four-man rotation.  Today the average team has four complete games per season.  Hitters today face pitchers who are better rested and more specialized.  A hitter typically will face, within the same game, both right-handed and left-handed pitchers who throw a wider selection of pitches than an individual would be able to master.

There are rule changes in baseball that also make comparisons difficult (but interesting) across eras.  When Yaz won the Triple Crown baseball was in the midst of a pitchers’ era; only four batters in the American League hit better than .300 and the following year Yaz repeated as the leading hitter at .301 with the second place hitter at .290.  Then the mound was lowered and hitters have gained on pitchers for most of the past 40 years.

Nonetheless, Cabrera’s accomplishment may not be repeated by another baseball player for decades.  Brees’ streak will probably continue throughout this season.  It is also likely that other top quarterbacks will challenge his record in the next decade as NFL teams rely more and more on the pass and record books continue to be rewritten every few years.

Aly Raisman was Robbed of a Bronze Medal: Why Gymnastics Scoring is Flawed

Aly Raisman of the US deserves the bronze medal for the women’s gymnastics all-around event.  After the scores were recorded from all four apparatus, Aly was tied with Aliya Mustafina of Russia.  The tie-breaker for this competition makes no sense.  The lowest score for each gymnast is dropped and then the total score over three (possibly different) events is compared.  This rule will often lead to the gymnast with a higher standard deviation in scores across apparatus winning the tie-breaker.

The competition is to determine which gymnast is better overall.  Holding constant the mean performance, the gymnast with the lower standard deviation in scores is the most consistent across apparatus.  That certainly sounds to me like a way to identify the better all-around gymnast.  On Thursday in London, Aly was a better all-around gymnast than Mustafina.  Aly’s standard deviation in scores across apparatus was 0.68 compared to Mustafina’s 0.90.

The International Gymnastics Federation and the International Olympic Committee should change the tie-breaker for future all-around events.  Even that won’t help Aly Raisman who was robbed of a bronze medal.  I hope the IOC comes to its senses and awards her the medal she deserves, but I am afraid that is unlikely to occur.

Starting Position Matters

Ryan Briscoe is in the pole position for today’s Indianapolis 500 and if history is our guide, he has the best chance of winning the race (14 of the past 50 winners started in the pole position).  Briscoe’s qualifying speed was less than two thousandths of one percent faster than James Hinchcliffe’s average speed.  Along the first two rows of cars (starting positions one through six) the difference in average qualifying speed between adjacent cars/drivers is about one tenth of one percent.  The following chart illustrates the relationship between the probability of winning the Indy 500 and starting race position.  Both the actual raw probability and the fitted probability from a logit model are included in the chart.

Among cars/drivers in the first two rows, each one place reduction in starting position is associated with a 25% reduction in the relative likelihood of winning the race.  For starters in rows three through five (positions 7 through 15), each one place reduction in starting position is associated with a 10% decline in the relative likelihood of winning the race.  A driver’s starting position has a much bigger impact on the likelihood of winning the race than one would expect from the small differences in the speed of the cars in the time trials.  Finally only 2 of 800 drivers in the bottom half of the field, ordered by qualifying times, won the Indianapolis 500 over the past 50 years (these cars/drivers are not included in the chart).

Starting positions for the Indy 500 have a disproportionate impact on the likelihood of success in the race.  A car with a poor starting position in today’s 500 mile race can still come from behind and win, but the odds are against the driver.  For an electorate that appears concerned that the economic playing field is not level, it is interesting to note that relatively small differences in initial conditions have a disproportionate impact on who wins and who loses the most iconic American race .

The NBA’s Small Market Ratings Problem

NBA Commissioner David Stern breathed a sigh of relief last night when the Miami Heat took charge in the fifth game of their playoff series with the Indiana Pacers.  The NBA already faces a ratings challenge in the Western Conference with the small market Oklahoma City Thunder and San Antonio Spurs.  After a protracted strike that shortened the season, compressed the schedule, and lowered the quality of play, the Association was hoping for a more appealing Finals match-up than they are likely to get.  The Thunder and Spurs are clearly the best teams in the West; the problem is they don’t have many fans.  An NBA Finals matchup against the Indiana Pacers would have produced historically low ratings.

Oklahoma City and San Antonio are the third and fourth smallest television markets in the Western Conference.  Only Memphis and New Orleans have smaller markets.  Before the season began there were 105 possible Western Conference Finals matchups, and 98 have larger combined markets than Spurs-Thunder.  There are fewer television households in Oklahoma City and San Antonio combined than there are in Minneapolis, alone.  The combined size of the television market for the Spurs and Thunder is 52% smaller than the median matchup (Minnesota Timberwolves – Denver Nuggets) and 81% less than the best matchup of the Lakers (or Clippers) and Dallas Mavericks.

The ratings of NBA Finals are directly related to the market size of the participating teams.  There have been 13 NBA Finals played since Michael Jordan left the Chicago Bulls.  The Spurs have played (and won) in four and the Los Angeles Lakers have played in seven of the Finals.  The ratings are 24% lower when the Spurs are in the Finals compared to the Lakers.

The challenges of a Spurs-Thunder matchup would have been compounded by an Indiana Pacers win last night.  Indianapolis is the sixth smallest market in the NBA and only Milwaukee is smaller in the Eastern Conference.  There has not been an NBA Finals from smaller markets than Oklahoma City and Indianapolis since the Fort Wayne Pistons played the Syracuse Nationals in 1955.  Conspiracy theorists were probably silenced last night because the game wasn’t close.  Nonetheless, the NBA and ABC, the network televising the Finals, are hoping that the midsize market Miami Heat (with considerable star power) or large market Celtics or 76ers prevail in the East.  Regardless of the Eastern Conference winner, however, the ratings for the NBA Finals are likely to be the lowest in history edging out the 2007 matchup between the Spurs and Lebron James’ Cleveland Cavaliers.

Home Ice Disadvantage in the NHL Playoffs (Updated April 27)

The first round of the NHL playoffs has ended and the lower seeded team won half of the eight series.  The higher seeded team won only 11 out of 27 games on their home ice compared to 14 of 21 games as the visiting team in this year’s NHL playoffs.  The higher seeded team has been 26% less likely to win a game on their home ice compared to an away game.  It appears, throughout the first round of the playoffs, that there is a home ice disadvantage in professional hockey.

In the previous six seasons there has been a smaller home ice advantage in first round playoff series compared to the regular season.  Between 2006 and 2011 the higher seeded team:

  • Won 57.5% of first round playoff games on their home ice
  • Won 54.6% of first-round playoff games on the road

If one examines the regular season games between teams that will eventually meet in the first round of the playoffs, between the 2005-2006 and 2010-2011 seasons, the higher seeded team:

  • Won 64.1% of regular season games on their home ice
  • Won 43.2% of regular season games on the road

In other words a home ice advantage of 17.9% in the regular season is reduced to 2.9% once the playoffs begin.

Why did favored home teams lose so many of their first round games in the 2012 NHL playoffs?  The higher seeded teams actually had a home ice disadvantage against their lower seeded opponents during the 2011-2012 regular season.  The higher seeded team won 53% of away games and only 44% of home games against their first-round playoff opponents during the regular season.  This 9% disadvantage to playing on a team’s home ice was even larger in the playoffs.

This year’s NHL playoffs have shown that a home ice advantage either doesn’t exist or is much less important than the advantage that home teams have in the playoffs in other professional team sports.

Mega Millions Frenzy

Mega Millions Frenzy

It’s difficult to make economic forecasts in uncharted territory but lottery officials have set a new standard for missing the mark.  Everyone realized that the Mega Millions jackpot would set a new record and that lottery sales would surge, but no one knew by how much.  Lottery officials consistently underestimated sales over the past few days.  The first announced Mega Millions jackpot was $460 million but reached $640 million by the end of the week.  Higher jackpots result from higher ticket sales, but more than two-thirds of new sales revenue goes to the states and consolation prizes.  This means that Mega Millions sales over the past three days were almost three times higher than suggested by the original advertised jackpot. 

Lottery officials seem to have badly underestimated Mega Millions frenzy.  Are lottery officials really that bad at forecasting sales?  Lottery officials are almost certainly required to make cautious forecasts so that the advertised jackpot rarely overstates the actual jackpot.  Do regulations requiring lottery officials to systematically underestimate jackpots depress sales?

Perhaps not, because conservative forecasts may promote sales as gradual increases in advertised jackpots are reported by the news media.  An unbiased forecast would be revised down as often as it is revised up, but a decrease in the advertised jackpot could provide negative publicity.  Would we have seen the same Mega Millions frenzy if the advertised jackpot had been revised down during the past few days?

The nature of Mega Millions arithmetic means that if a jackpot has accumulated over time, as we observed from January 24 until this week, increases in the size of the advertised jackpot during the week is bad news for the customer.  This is because rising sales during the week increase the jackpot by far less than they increase the chance of a winning ticket sharing the jackpot.

If the actual jackpot had been $460 million as originally “forecast” by officials, the winning ticket would have been expected to generate a payout about 20% more than Friday’s winning tickets from a much larger jackpot.  With fewer sales during the week the most likely outcomes are that a winning ticket shares the jackpot with one other contestant or receives the prize alone.  As sales grew during the past few days it became much more likely that the jackpot would be split three ways, which is what happened last night.

Unless lottery officials change the Mega Millions rules we won’t see a jackpot this large for at least five years.  The accumulated jackpot grew to record levels because over 800 million tickets didn’t match the winning numbers over 18 previous drawings.  The odds of this happening are less than one in one hundred.  Over the past four years the typical jackpot was awarded after accumulating over 8 weeks, and Friday’s jackpot was just the third that had accumulated for more than 15 drawings.

When the next record jackpot occurs players should be skeptical of the initial jackpot announcements and expect the advertised amounts to increase as the next Mega Millions frenzy appears to gain momentum.  When this happens players should also remember that jackpot increases during a week of record Mega Millions sales actually decrease the expected payout from a winning ticket.

Why a Mega Millions Ticket is a Good Bet, But… (Revised 03.30.2012)

The Mega Millions multi-state lottery is now advertising a jackpot of $640 million for tonight’s drawing. The cash value of the jackpot is $460 million, a more accurate estimate of the present value of the prize. The jackpot has accumulated because there has been no winner for 18 consecutive drawings – since January 24th. This means that those of us who buy Mega Millions tickets for the first time this week have been subsidized by the millions of players who contributed to the jackpot over the past two months.

The probability of correctly selecting all six numbers for Friday’s drawing is one in 175 million. It is likely that there will be multiple jackpot winners. Even after taking this into account, the expected value of a one dollar Mega Millions ticket is about 90 cents.

About 18 cents of every dollar collected from Mega Millions sales are used to pay consolation prizes. Another 32 cents of each dollar goes into the cash jackpot. The cash value of Friday’s drawing is estimated to be about $200 million higher than in Tuesday’s drawing. In other words lottery officials expect about 625 million lottery tickets to be sold between now and Friday.

If lottery officials are correct there is a 96% chance that someone will pick the winning numbers on Friday, and the expected number of winners, conditional on the jackpot being awarded, is about 3. This means that even after splitting the jackpot, the expected value of a Mega Millions ticket is about 90 cents (including 18 cents in expected consolation prizes).

In general state sponsored lottery games have low expected values for players, relative to other forms of gaming. Lotteries pay out about half of their proceeds in prizes, on average. The rollover nature of the jackpot in state lotteries means that a lottery ticket will have a high expected value on occasion. Until the recent surge in sales, this week would have been the highest expected value for a Mega Millions ticket in the history of the game.

Finally, because the jackpot will be taxed at a fairly high rate a Mega Millions ticket is not a good investment, as Ben Casselman of the Wall Street Journal has noted. After taxes the expected value of a Mega Millions ticket is now much less than $1, unless the top marginal tax rate falls substantially over the next twenty years, which is doubtful. However most gambles with long odds, such as a super trifecta bet at the racetrack, have a pre-tax expected value that is lower than the cost of the wager. So the Mega Millions is an unusually good bet for people who enjoy gambling small amounts at long odds, but it would be a bad investment.

The Super Bowl and the Oscars

For the second year in a row, 111 million Americans watched the Super Bowl.  Prime time television has been dominated by reality shows, of late, with American Idol holding the top spot among network shows six years running.  The Super Bowl has become the reality television event, surpassing the ratings of all other broadcasts year after year.

Network television ratings have declined steadily over the past few decades as networks face increased competition from cable, satellite, and online videos.  The Super Bowl has defied this trend.  There was a time when the Academy Awards show was the television event of the year.  The first few Super Bowls drew about the same audience as the Oscars.  Since then it hasn’t even been close.

As I watched Sunday’s game I realized that the Super Bowl, as an event, has taken the place of the Academy Awards in many ways.  The comic relief previously supplied by Bob Hope, Johnny Carson, or Billy Crystal has been replaced by amusing commercials, some starring Jay Leno and Jerry Seinfeld.  An over-produced half-time show has replaced the song and dance routines that opened each Oscar telecast.  Although we saw fewer movie stars Sunday than at the Dorothy Chandler Pavilion, Matthew Broderick and Clint Eastwood were trying to sell us cars.

For many years the game didn’t live up to the hype.  Between 1983 and 2001 only 2 of 19 Super Bowls were decided by less than a touchdown, and the audience did not grow.  Seven of the last 11 games were decided by six points or less, and the audience has grown by 30%.  The audience for Sunday’s game, which came down to the last play, peaked in the fourth quarter at 117 million.  As long as the games remain competitive the Super Bowl will likely be the only network telecast with an audience that increases each year.

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