Grappling with Vegas math

Image of math teacher in front of chalk board One thing which has long puzzled and troubled me is the manner in which games of chance pimp their odds. If, for example, we toss a coin in the air there's a 50% chance it will land "head's up" and a 50% chance it will land "head's down". If the coin is a Loonie, Canada's $1 coin of the realm, there's a 100% chance it will land "squawking buzzard up". For those who may not know, one side of the loonie is duly embossed with a loon while the other side is graced by Lizzy Saxe-Coburg-Gotha. Saxe-Coburg-Gotha is not, as her obscure namesake might suggest, a renowned German porn queen.

If a coin, tossed in the air 10,000 times landed "head's up" every single time, how would you bet the coin would land on the 10,001st coin toss? If you answered an unequivocal "TAILS" then I'd suggest you've just fallen victim to, what I affectionately refer to as, the gambler's falicy.

Of course beancounters and statisticains employ complex mathematical formulae and analyses to calculate probabilities and the games are always skewed in favour of the house. The roulette wheel, for example, has an even number of black and red outcomes but it also has one or two green outcomes (0 & 00) which represent the "house edge". Even something as seemingly innocuous as insurance is nothing short of a game of chance; a wager based on a complex set of confounding variables. Naturally, the premiums are adjusted according to risk assessments and the odds always, given a large sample, favour the insurer, not the insured. It's almost ironic that life insurance clients are betting they will die prematurely whereas insurers are hedging their bets that the client won't be conversing with Elvis any time soon.

BEST ODDS EVER - 1 in 3; or at least that's what the promo on the website asserts in the Heart & Stroke Lottery. In this particular lottery 250,000 tickets are printed and a total of 71,653 prizes are awarded. The FAQs on the website assert that "the lottery industry practice is to calculate the odds by taking the number of tickets sold, divided by the number of prizes available. That is 250,000 tickets / 71,653 prizes = 3". But are these truly the odds of winning?

Mercifully, I don't subscribe to Vegas math so I'm quick to toss the calculated lottery industry standard odds of 1 in 3 right out the window and adopt the more cynical "Big Banana Bob standard" which calculate my odds from a more reasoned perspective. The Big Banana Bob standard, incidentally, calculates that I have 71,653 opportunities at winning a prize, each with the odds of 1 in 250,000.

I've been trying to understand precisely how the industry standard calculates their odds and as near as I can figure out, without engaging in too much arithmetic hyperbole, they simply view odds in games of chance somehow as an additive process. In other words, 1/250,000 + 1/250,000 + 1/250,000 ..... + 1/250,0000 (71,653 times) = 71,653/250,00 = 0.2866. It would appear this 28.66% prize/ticket ratio is being presented as 1 in 3 odds of winning a prize. If, however, games of chance were additive, would it not logically follow that 1/3 + 1/3 + 1/3 = 3/3 = 1.00? In other words, if, as the industry standard suggests, your odds are additive, then purchasing 3 tickets at 1 in 3 odds should guarantee a prize winning ticket, n'est ce pas?

So, if I buy three tickets, are the odds greater that all three will be among the 71,653 winning tickets? Or are the odds greater that all 3 will be among the 178,347 losing tickets? How many tickets do I have to buy to guarantee one of them will be a winning ticket? Whereas the industry standard seems to suggest purchasing 3 tickets might ensue a prize, the Big Banana Bob standard states, quite unequivocally, that if you buy 178,348 tickets you are absolutely, without a doubt, going to win a prize.

I've concluded that there's no fiscally viable means I could take to guarantee I will win a prize in these sorts of lotteries and that opting out is the only way I can ensure I do not purchase a losing ticket. I can easily reconcile my feelings that the noble purpose of such lotteries is to raise funds for worthwhile causes and, to this end, I can simply send them a donation instead of buying their lottery ticket. The donation, incidentally, is tax deductable whereas the lottery ticket is not. Moreover, I'd wager two bottles of my home-made swill that their take-home portion from a straight cash donation is higher than an equivalent amount spent on one of their lottery tickets.

That reminds me.... it's December so I should fire off my annual contribution to the War Amps. No fancy lottery to concern myself with and they send me a regular supply of sticky return address labels I can use on my personal correspondence. Not sure what the odds are that I'll ever benefit from their key tag programme but I'm thinking it's somewhere in the same neighbourhood as being hit by a Soviet space probe plummeting out of the sky. Incidentally, the War Amps have returned over one million sets of keys since 1946 and the Soviet's $170m radioactive Probos will hit the earth sometime next month.

Submitted by "Big Banana" Bob Loblaw, 22 December 2011