## A Periodic Look at the California Lottery

What are the odds of winning the California Lottery? I am talking about the winning of $1 million or more (the kind of winning that is a game changer in one’s personal life). How often are these million-dollar tickets won? Ten months ago I estimated that the odds of winning$1 million or more in the California Lottery were one in 36 million (see Taking another look at the California Lottery). The data were based on data from California Lottery that I obtained in November 2010 (see Shining a light on the California Lottery). Nothing happened in the last ten months indicates that the odds of winning has fundamentally changed.

Just to confirm, I count the number of winning million-dollar winning tickets as of today (August 30, 2011). This is done at the website of the California Lottery. The data are not readily available. I have to search at this site. I count the tickets by searching one county at a time (there are 58 counties in the state). The result: since the inception of the California Lottery in 1985, there are only 257 tickets that paid out $1 million or more (an increase of 10 winning tickets over 10 months ago). So in its 26-year history, there are only about 260 winning tickets, about 10 per year. The increase of 10 tickets in the last 10 months also confirms the average of 10 winning tickets per year. With the increase of 10 more winning tickets, the odds are actually a little higher, about one in 36.7 million, but still not fundamentally different from 1 in 36 million. The mantra of many lotto players is that you have to buy a ticket in order to win. That is so true. You have to get in the game to have a chance to win, even though the chance of winning is infinitesimally small. On average it takes the purchase of about 36 million tickets to support one winning ticket. Still dreaming of winning big? Advertisements ## How many lottery winners are there in a year? I have been wanting to answer the questions in the title. I found that statistics on lottery winning is hard to come by. Even when the state lottery commissions are required by law to made the information public, they tend to bury the information and you have to do work to dig it up. I have strong indication that on an annual basis, winning tickets that pay out one million dollars or more only number in the hundreds. In contrast, there were 37,261 people killed in motor vehicle crashes in 2008 in the United States (see the report from the National Highway Traffic Satety Administration). So if you are passionate about winning various state lotteries, it makes sense to be passionate about not winning the negative lottery of fatality in a motor vehicle crash too. As of November 2010, there were only 247 winning tickets paying one million dollars or more (see the previous post with this discussion). To get this information, I had to look up the winning tickets in each of the 58 California counties in the official site of CalLottery. So about 10 people are made millionaires by CalLottery each year (since its inception 25 years ago). The state of Iowa is more forthcoming. The official site of the Iowa Lottery actually had a press release listing out the stats. The number of Iowa Lottery tickets that have won prizes of$1 million or more (through August 2010) is 110. Once again in the 25 years history of the Iowa Lottery, only 110 people were made millionaires, on average 4.4 per year. For the Iowa Lottery, the odds for winning $100,000 or more are better for sure (1089 winnings so far in 25 years) but the odds are still small. The state lotteries are in the business of selling dreams. I suspect that they do not want to provide a picture reflecting the true odds of winning big. With all the state lottery commisions across the United States combined, I cannot see how the number of winning tickets ($1 million or more in each one) in one year can be in the thousands. If someone is forking over hard earned cash each week to play the lottery in the hope of winning big, it also makes sense to pay attention to traffic safety in the hope of not winning the negative lottery of death in a car crash.

Update (12/20/2017). A recent blog post answer the same question for Powerball. The winning of the Powerball jackpot occurs much less frequently than used to be. This is a cynical ploy to drive ticket sales. You can read it here. The blog post gives the link to the blog post in a companion blog that gives the graphics and analysis.

## Taking another look at the California Lottery

What are the odds of winning the lottery? If the lottery is a 49/6 game (i.e. choosing 6 numbers out of 49 numbers), the odds are 1 in 14 million (one in 13,983,816 to be precise). I would like to show you that for the California Lottery, the odds of winning a jackpot of $1 million or more are 1 in 36 million. Let me show you how I arrive at this estimate. Statistics about winning tickets are available from the official website of the California Lottery. But one has to do some digging to get the data (I searched at the Lucky Retailer Search). There are 58 counties in California. I simply searched for the 58 counties one by one. Only 28 of the counties had winning tickets. Since the inception of the California Lottery 25 years ago, there were 247 winning tickets as of November 1, 2010. Here’s the summary information. All of these 247 tickets paid out$1 million or more. The largest jackpot was $110 million. The earliest winning ticket was on 3/21/1987, bought from a retailer in Imperial County. The most recent one was on 10/9/2010, bought from a retailer in Ventura County. The sum of all the winning amounts for these 247 tickets was$4,535,519,264 (about $4.5 billion). Thus each winning ticket prize was, on average,$18,362,426 (about $18 million). So there are about 250 winning tickets that paid$1 million or more in the 25 years of history of the California Lottery. On average, there were about 10 winners a year. If you do not think that the odds are infinitesimally small, read on.

By law, the California Lottery has to pay out at least 50% of the revenue as winning. The total winning amount for these 247 tickets was $4.5 billion. This implies that the$4.5 billion in winnings was paid out from the sales of $9 billion worth of tickets (equivalently 9 billion tickets since the ticket price was$1).

So out of 9 billion tickets bought, there were about 250 winners. Thus the odds of winning are 250 in 9 billion or 1 in 0.036 billion (9/250=0.036). The odds of 1 in 36 million followed from the following translation.

$\text{1 billion = 1,000,000,000 (1 followed by 9 zeros)}$
$\text{0.036 billion} = 0.036 \times 1,000,000,000=36,000,000 \text{ (36 million)}$

Of course, the California Lottery will never tell you that the odds of winning the big prizes are 1 in 36 million. One has to dig to find the information like I did. In fact, in the same page where I did the digging, they claim that “Since 1985, the California Lottery has distributed more than $27 Billion in winnings (including annuitized) with more than 2,842,467,062 winning tickets sold! They claim that there were 2,842,467,062 (2.8 billion) winning tickets since 1985. How does this number squared with the 247 tickets that I found? I wrote about this point in a previous post called Shining a light on the California Lottery. Except for 247 tickets, these tickets paid out small prizes (on average just under$10). Their information is correct but can give the impression that there are many millionaires running around (could be as many as half the world’s population)!

The usual refrain of many lotto players is that you have to buy a ticket in order to win. Winning is desirable for sure. In the case of mega lotto jackpot such as the games of Mega Million and SuperLottoPlus in the California Lottery, you have to buy millions of tickets before you have a realistic chance of winning (could very well be 36 million tickets). If you treat the game of lottery as a money making opportunity or a way to become an instant millionaire, you better count the cost. Some play the lottery for its entertainment value and the excitement. If you spend a small sum every week buying tickets for the huge jackpots, the entertainment value is about the only benefit you will receive from playing.

## Shining a light on the California Lottery

In the back of my mind, winning the lottery means becoming a millionaire (or better). Thus I find the following statements found in the website of the California Lottery very interesting.

Since 1985, the California Lottery has distributed more than $27 Billion in winnings (including annuitized) with more than 2,842,467,062 winning tickets sold! The amount of$27 billion is a lot of money. This amount ought to be reassuring to anyone dreaming of winning big (think sitting in a beach chair holding a martini in a posh beach resort in the Caribbean). The number of winning tickets 2,842,467,062 is a huge number too. If I keep playing, could I be joining this huge legion of winners?

I noticed something else. Why is the total amount of winnings stated in a nice round number while the total number of winning tickets is not? Note that both figures are not meant to be exact amounts (more than $27 billion and more than 2,842,467,062 winning tickets). Why not express the number of winning tickets in a nice round number too? Could it be that this is intentional? Could it be that the display of the number 2,842,467,062 is designed to be fantasy inducing? I do not know for sure. But I have my suspicion. Understanding how to read large numbers will clear things up. One million is 1,000,000 (one followed by 6 zeros). Note that one million is 1000 times of 1000. Putting it another way, if you receive$1000 from each of 1000 people, you become a millionaire.

One billion is 1000 times of one million (1,000,000,000 or one followed by 9 zeros). If you have $1 billion in wealth and you only spend$1 million a year, it will take you 1000 years to deplete your wealth! Of course, in this scenario, we are not taking the time value of money into account (but that is another story).

So the number of winning tickets for the California Lottery since 1985 is 2,842,467,062 (about 2.8 billion tickets). Interestingly, this means that the average winning amount per ticket is slightly under $10! This implies that most of the 2.8 billion tickets are for small prizes (way smaller than$1 million).

So how many lottery prizes of $1 million or more were won by players in the California Lottery in its 25-year existence? Fortunately, the data are available in the official website of the California Lottery, just that they are not conveniently summarized. I had to search for them county by county (there are 58 counties in California). I searched the winning tickets by county and I found a total of 247 winning tickets, all in the amount of$1 million or more. These 247 tickets amount to $4,535,519,264 (or$4.5 billion). So more than $22.5 billion (=27-4.5) in winnings are for smaller prizes (e.g. a few hundreds to tens of thousands in dollars). Out of 25 years of history in the California Lottery, there are only about 250 winning tickets with$1 million or more in winning. On average, there are about 10 such winning tickets a year. So winning a small prize may have good odds (about 2.8 billion instances of small winning so far). But winning a huge jackpot in the California Lottery, one that you normally think of as setting you up for life, had only happened 250 times so far.

The California Lottery is in the business of selling dreams. It seems that fuzzy numbers help keep dreams alive. Interestingly, a large number such as 2,842,467,062 was transformed into a fuzzy number by not rounding it.

Even if I did not dig up numbers from the official website, I can still get a sense that there is only a small number of winning tickets worth $1 million (or more). We can compare 2,842,467,062 with the sizes of the population in California, the United States and China. The number 2,842,467,062 is almost 77 times the population of California (36.9 million in 2009), and is over 9 times the population of the United States (307 million in 2009). The population of China is 1.3 billion (in mid 2008). The number 2,842,467,062 is over twice the population of China. Imagine that the number of millionaires created as a result of playing the California Lottery is twice the China population! If true, California would truly be a “Golden State”! What can one get from buying a$1 lottery ticket? Not sure what one can get other than a chance to fantasize (a form of cheap entertainment I suppose). The fact speaks for itself. Of the tens of billions of tickets bought in the 25-year history of the California Lottery, there were only 247 winning tickets that paid out $1 million or more. ## The lottery of drunk driving fatality Several years ago, I read of a news account of a mother driving with her infant son in a minivan. A drunk driver was driving on the same freeway but in the wrong direction and collided with the minivan. The mother died upon impact. Amazingly the infant survived. News accounts like this one are always heart wrenching. I always think of dying from a crash involving an alcohol-impaired driver is a lottery. It is a negative lottery for sure since no one would want to win it. Los Angeles Angels pitcher Nick Adenhart and two of his friends were the unfortunate “winners” of this negative lottery in 2009. What are the odds of winning this negative lottery? How often is this negative lottery “won”? Let’s compare the number of drunk driving deaths with the number of winning Lotto tickets that pay out$1 million or more. The comparison is done on an annual basis and as a rate per certain number of minutes. It turns out that winning this negative lottery is far more frequent than winning the Lotto. Driving under the influence of alcohol is a serious problem. The numerical data presented here bear that out.

Drivers are considered to be alcohol-impaired when their blood alcohol concentration (BAC) is 0.08 grams per deciliter (g/dL) or higher. Thus any fatality occurring in a crash involving a driver with a BAC of 0.08 or higher is considered to be an alcohol-impaired driving fatality. According to a report from the Center for Disease Control and Prevention (CDC), one alcohol-impaired driving fatality occurs every 45 minutes (see note 1 at the end of the post). This rate of occurrence is derived from the fact that in 2008, 11,773 people were killed in alcohol-impaired driving crashes.

What is the total number of Lotto winners in a year? Here, we only focus on the winning prizes of $1 million or more. Would the number of Lotto winners approaches 11,773 in a year? I do not think so. I hunted for data from the website of the California Lotto, I found that as of today’s date, there are only 247 winning tickets that pay out$1 million or more since the inception of the California state lotto in 1985. So in 25 years, there are only about 250 winners in the state of California. Thus on average there are only about 10 “million dollar plus” Lotto winners a year in California.

Multiplying across the 50 states in the United States, the total number of “million dollar plus” Lotto winners should be no more than 500 in a year. This should be a pretty conservative estimate since California is the largest state in the country (with the largest population). Not all 50 states in the country have Lotto. Some states do not have their own Lotto and are part of a multi-state Lotto. So 500 is a good upper bound on the total number of “million dollar plus” Lotto winners in a year.

With 500 winners a year, there is one “million dollar plus” Lotto winner every 1050 minutes (see note 2). Or one winner in every 17.5 hours (see note 2). Thus alcohol-impaired driving deaths occur at least 23 times more frequently (one death per 45 minutes vs. one per at least 1050 minutes).

Of the 11,773 alcohol-impaired driving deaths in 2008, how many of them were the drunk drivers? According to a report from National Highway Traffic Safety Administration (NHTSA), an agency within the Department of Transportation, we have the following breakdown of the drunk driving fatality in 2008.

$\displaystyle \begin{pmatrix} \text{Role}&\text{Number}&\text{Percent of Total} \\{\text{ }}&\text{ }&\text{ } \\\text{Driver With BAC=0.08+}&8,027&\text{68 percent} \\\text{Passenger Riding w/Driver With BAC=0.08+}&1,875&\text{16 percent} \\\text{Occupants of Other vehicles}&1,179&\text{10 percent} \\\text{Nonoccupants}&692&\text{6 percent} \\{\text{ }}&\text{ }&\text{ } \\{\text{Total Fatalities}}&11,773&\text{100 percent}\end{pmatrix}$

Most of the deaths were self-inflicted (68%). According to the same report, 216 of the 1,875 deaths in the above table were children. The group of 2,087 (=216+1,179+692) were truly victims and their deaths were senseless. Angels pictcher Nick Adenhart and the mother mentioned at the beginning belong to this group. They did not do anything wrong and were just in the wrong place at the wrong time.

If we just focus on this group of children riding with the drunk drivers and the occupants of other vehicles and nonoccupants, the rate of occurence is still pretty high (one fatality every 252 minutes; see note 3). That comes up to be about one fatality every 4.2 hours (see note 3). Focusing on this group alone, the rate of drunk driving fatalities is over 4 times higher than winning the Lotto (one death per 252 minutes vs. one winner per at least 1050 minutes).

Eliminating the category of fatality represented by these 2,087 senseless deaths would drive down the rate of fatality and would go a long way to address the problem of drunk driving. Though this would be only modest improvement, it would be a step in the right direction.

*************

Note 1
To derive the rate of one death per 45 minutes, we need to calculate the total number of minutes in a year. There are 365 x 24 x 60 = 525,600 minutes in a year. Then divide 525,600 by 11,773 to obtain 44.64 minutes = 45 minuties.

We can get a perspective of this calculation by looking at an example of taking an exam. For example, if you have two hours (120 minutes) to take an exam and the exam has 10 problems, then on average you have 12 minutes to work one problem. Thus if you can work one problem per 12 minutes, you can expect to finish the exam in the allotted time.

Back to the alcohol impaired driving situation, there are 525,600 minutes in a year and there are 11,773 events. Thus on average there are 45 minutes allotted for each event.

$\displaystyle \frac{365 \times 24 \times 60}{11,773}=44.64 = 45 \text{ minutes}$

Note 2

$\displaystyle \frac{365 \times 24 \times 60}{500}=1051.2 = 1050 \text{ minutes}$

$\displaystyle \frac{1050}{60}= 17.5 \text{ hours}$

Note 3

$\displaystyle \frac{365 \times 24 \times 60}{2087}= 252 \text{ minutes}$

$\displaystyle \frac{252}{60}= 4.2 \text{ hours}$