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May 2005

Powerball Analysis

After originally posting this, powerball changed the odds and payouts. Instead of drawing the group of 5 from a set of 53, they draw from a set of 55. I assumed that this would be simply a reduction in the expected value of a ticket, but they also tweaked the payouts for 2 of the non grand prize winners, and the result was that the expected value of a non grand prize winner actually increased, from 17 cents to 20 cents.

Some of the increase in expected value was driven by the increased odds of 'winning' by not getting correct numbers in the group of 5. This is a very small effect, however, since the odds only actually increased for the nil case of getting no correct numbers.

This tells me that powerball revenue is affected as much by psychology as statistics. The net effect of the changes was to dramatically increase the odds against winning the grand prize, thus driving the average grand prize upward, thus increasing ticket sales. And since the expected value is still very low, increased ticket sales translates directly to increased revenue. In fact, a 4% increase in sales would offset the increased expected value.

Here is the new analysis, followed by the previous analysis.

Five balls are drawn from a drum with balls numbered 1-55. One ball is drawn from a drum with balls numbered 1-42.

The total number of combinations is

total = c(5,55) * 42 = 146,107,962

In the table above, payout is the amount that would be paid out for each type of prize assuming that all combinations were selected. The total payout for all combinations other than the winner is $20,888,592, or just under one sixth the total amount.

The following table shows statistical expected value of a $1 ticket for a range of grand prizes. I have extended the table to reflect the fact that the grand-prize nominal amount is over twice as high as the actual amount you would receive (before taxes) if you take the one-time payout option.

Ignoring the grand prize, the expected value of a $1 ticket is 19.7 cents. And the odds of winning the grand prize are so vanishingly small that it is reasonable to ignore it in the expected value calculation. Thus, you can expect that you will on average get back less than $10 for each $50 that you spend - not a good value proposition.

The marketing of powerball, and of state-run lotteries in general, seems to me to verge on fraudulent in 2 ways:

1. The stated prize value is misleading at best. The nominal payout is paid out over a 30 year period, rendering its net present value far less than the nominal value. The actual present value works out to about 47% of the stated prize amount. This fact should be prominently stated in all marketing material for the lotteries.
2. It is unreasonable to expect that consumers will engage in a laborious statistical analysis of the odds, payouts, and expected values of lottery games. Lottery game purveyors should be required to distribute an analysis, similar to the one taken above, to allow consumers to make better informed decisions about lottery purchases. Failing to do so simply exploits the inability of most people to reason effectively about very large or very small numbers.

I'm sure it is very clever to say that "lotteries are a tax on the (mathematically) stupid." But as a society isn't it our responsibility to provide the tools for people to make informed choices? In fact, haven't we accepted that responsibility in nearly every other area of our lives? We insist on truth in advertising for everything from corn chips to cars. But we exempt the lottery from that same requirement.

Powerball Analysis (pre August 2005)

Five balls are drawn from a drum with balls numbered 1-53. One ball is drawn from a drum with balls numbered 1-42.

The total number of combinations is

total = c(5,53) * 42 = 120,526,770

In the table above, payout is the amount that would be paid out for each type of prize assuming that all combinations were selected. The total payout for all combinations other than the winner is $20,888,592, or just under one sixth the total amount.

The following table shows statistical expected value of a $1 ticket for a range of grand prizes:

Ignoring the grand prize, the expected value of a $1 ticket is 17.3 cents. And the odds of winning the grand prize are so vanishingly small that it is reasonable to ignore it in the expected value calculation. Thus, you can expect that you will on average get back about $10 for each $58 that you spend - not a good value proposition.

Perpetual War

For two and a half years I have been trying to understand the motivations behind the Bush administration's determination to attack Iraq. The idea that the attack was motivated simply by a desire to secure oil resources seemed facile - after all, Bush and his cronies have plenty of ways to make money from oil without securing the Iraqi oil fields. The Afghan pipeline boondoggle is just one example; opening up the Alaska north slope wilderness is another.

The unprovoked attack on Iraq could not have been motivated, or justified, by the need to combat terrorism : Iraq under Saddam was no friend of terrorists. Saddam's mission was to increase his own power, not to aid Islamic extremists whom he regarded with disdain and contempt. Though Bush tried to wave the anti-terrorism flag as part of the initial justification of the war, he quickly backed off once the absurdity of the argument was apparent.

The justification came down to the issue of 'weapons of mass destruction'. We know now, and suspected then, that the 'smoking gun / mushroom cloud' argument was trumped up and based on intelligence findings that had been guided if not mandated by the Bush administration. The New York Review ran an article (I'll see if I can find a link) in 2003 showing that of the 27 claims about Iraqi WMD made by Colin Powell to the United Nations in February 2002, all were either entirely or mostly false. Bush knew that the WMD claims were bogus.

So - what was the motivation?

The recent passage of the 'Real ID' legislation, buried in an $80 billion 'emergency spending bill' to continue financing the Iraq debacle provides the answer. Perpetual war allows the Bush administration to have its way with America. Congressmen who normally would see the Patriot Act and Real ID as the threats to liberty that they are, and who would otherwise oppose them, are blackmailed into support in order to avoid accusations of 'not supporting the troops' or even of being 'unpatriotic'.

The Bush administration, and the radical right-wing generally, seek to transform America into an oligopoly ruled by the very wealthy. Every domestic program pushed by the Bush administration has had that aim : elimination of the estate tax, tax cuts benefitting the wealthy at the expense of working people, gutting of Social Security, passage of the draconian anti-bankruptcy bill. The Patriot Act, predating the unprovoked attack on Iraq but justified by the perpetual 'war' declared on terrorism, strengthens federal police powers in ways that would have been unimaginable and repugnant to the authors of our constitution. But such an act is necessary if you plan to take power from the people and hand it to a small group of the super-wealthy.

Bush's fiscal policy has been one of bankrupting the government. Why would a conservative do that? Aren't conservatives characterized by their fiscal responsibility? The answer is simple: the best way to eliminate social programs is to bankrupt the government and then make the argument that we simply can't afford those programs anymore. And that, of course, is exactly what Bush is doing.

We have entered a new period in American history. We are no longer the land of the free, and the rest of the world knows it though most Americans can't see it. We are no longer the shining beacon of liberty and opportunity. We are a nation of sheep being led by wolves. Always before, the question for people outside the United States was how to get in. Now the question I ask myself is - how do I get out?

The answer, of course, is that this is my country and I shouldn't have to get out. Our government has been of, by, and for the people, and it can be again. But we will have to fight to make it so. We must confront every lie, expose every act of corruption and cronyism, resist every piece of legislation that threatens our way of life.

But it is not enough to merely resist : we must define our principles and take every opportunity to advance them. We lost in 2004 because we were afraid to say what we really thought. The democratic party nominated a man who did not have the moral courage to stand up to the Bush administration when it counted. He voted to fund the Iraq adventure and thereby lost all credibility. We can do better, and we must if our country is not to be dragged further into the mud.