It's like a drug...

...this computer, but I'm waiting for Lee to come home from school, too, so I have a ready excuse and rationalization at hand.

I did a little more tweaking of my page configuration, adding/moving the link to `About LiveJournal.com' to the navigation area of my layout. I know there's a link to my home page at the bottom of the LJ output, but I'm wondering if it would be outré to move the thing up to the top of the page. Decisions, decisions.

Greetings, by the way, to new LJ friend bryn!

And now, dear friends, if's off to Dreamland, preferably not "a route obscure and lonely."

Cheers...

What're the odds?

Then again, who cares? As an old teacher of mine once remarked: "The odds of getting mauled by a Bengal tiger at high noon at the intersection of Broadway and 57th Street in Manhattan are about fifteen quintillion to one. But if it does happen, it will ruin your day."

I was talking to Drew on the phone the other day and he was telling me of his leisure plans for the near future. Considering the kid is working three jobs right now, I'm wondering where he gets the time to even think about taking time off (much less actually taking the time). As we're talking, I'm unconsciously running my eye down a list of items in a shipping manifest I'm reviewing for work.

At one point, Drew mentions that he's fixing to go cook dinner for himself and his girlfriend. As he asks this, I'm skimming to the bottom of a page, and as I turn to the next page, I ask "What are you cooking?" As I finish turning the page and focus my eye on the top line, Drew says "Fish."

The top line of the page in front of my face read: "Fish."

What're the odds? Who cares?

Then today, on the way home, I suddenly decided to stop at a store called Half Price Books, to see if they had any cool new computer books, any old novels by Mickey Spillane, or a CD of Richard Rodgers' Victory At Sea. I've had my eye out for this CD for some time; I'd even hit Napster for individual cuts from the album from time to time, with little luck. I'd have ordered a copy from Amazon a long time ago, except that, well...I can't explain why. I've ordered lots of stuff online (including toothpaste, energy bars, and dog food, if you can believe that), but I can't bring myself to order books or CDs from Amazon. Go figure.

As you might expect, the CD rack at this store reflects the music of the moment. Lots of rock, techno, soul, and hip-hop, and more New Age disks than classical ones. So I walk up to the rack and kind of run my eyes cursorily over the titles, only to find a copy of Victory At Sea lying on top of the rack, sitting on top of a bunch of New Age titles jammed edgewise in the rack. The jewel case is lying there in all its glory, as if someone put it out for me.

What're the odds? Who cares?

I am debating whether or not to go out and buy a lottery ticket. :^)

Cheers...

P.S. Greetings to new LJ friends pixelpusher and teferi. Welcome.

Lottery...feh!

Here in Texas, the state lottery folks have added four numbers to the fifty already in play to make the game "more interesting," or some other similar spin. They say it'll make the game more exciting because there'll be bigger jackpots to play for, and more often, too.

Of course, nobody among them is willing to demonstrate the truth of this statement mathematically, because that might affect sales. It may sprout a lynch mob or two, as well.

To wit, in a game that consists of guessing six out of fifty numbers, where the order of the numbers doesn't count, there are

50 x 49 x 48 x 47 x 46 x 45 = 11,441,304,000
different ways the game can turn out. That's over 11 billion possibilities.

Of these, there are 6 * 5 * 4 * 3 * 2 * 1 = 720 ways to arrange the winning numbers. (If the winning numbers were 1-2-3-4-5-6, you could win whether the numbers were picked in that order, or in any of the other 719 possible ways.)

Hold that thought while I digress:

There is a concept in probability theory called expected value. Basically, you multiply your payoff - what you can win - times the chances of your winning and that's your expected value. If the expected value is less than your bet, the game is - all things being equal - not worth playing because over the long run, you'll lose money. If the expected value is larger than your bet, the numbers say you should play the game, because in the long run you'll win more than you lose. Example: You play a game where you bet $1 and roll one die. If you roll a 6, you win back your dollar and another dollar. Is this a good deal? Answer: No. Your bet is $1. Your "jackpot" is $2. The chance of rolling a 6 - and we assume the die is honest - is 1/6. If we do the multiplication ($2 x 1/6), we find the expected value of any roll is $0.33. If you play the game a thousand times, you'll very likely lose around $667. What fuels gamblers everywhere is the prospect of having the win occur early in play. If you bet on a Las Vegas roulette wheel and hit a single number on the first spin, you'll be ahead. If you keep playing, though, the percentages will catch up with you and you'll end up paying the difference between what the house offers and what the wheel says. End of lecture.

If you consider that the odds of winning are 720 divided by 11,441,304,000 (or one in 15,890,700, a Real Small Number), and if you assume that the lottery you're playing for is $4 million, then the expected value of any ticket is:

720 * $4,000,000 / 11,441,304,000 = $0.2517
which is just over a quarter. This means that, for every bet of a dollar, you can expect a "return" of twenty-five cents, assuming the payoff is $4 million. (This analysis, by the way, does not take into account any of the lesser prizes in a state lottery, for example, getting four out of six numbers. The existence of those prizes does not materially change this analysis.)

So how large should the jackpot be for the bet to be "reasonable" by statistical standards? Take a look at the odds: As soon as the jackpot rises above $15,890,700, the percentages turn in your favor in the long run.

When you add four numbers to the pot, the number of possible outcomes increases to:

54 x 53 x 52 x 51 x 50 x 49 = 18,595,558,800.
That's a whopping 62% increase in the number of possible outcomes. With this set of balls, the expected value of a ticket for a $4 million jackpot drops to under 16 cents. (And yes, the pun is intended!)

The size of the "break even" jackpot - as far as expected value is concerned - is now $25,827,165.

It's only natural that pots will get larger, and that those big pots will occur more often, since it's now 62% more difficult to hit the right numbers!

State lotteries have been described as "a tax on the stupid," since - as I've attempted to show - if you played a dollar a week, you'd be lots better off putting the dollar in a bank at zero percent interest. At least after 20 years, you'd have a guaranteed thousand or so dollars in the bank. Playing the lottery, you'd all-but-certainly have a goose egg, and a lot of bookmarks.

Cheers...

Note: This post was revised on 9/20/00 to correct some flawed analysis.