A lottery math problem

The MegaMillion Lotto Jackpot is now $237 Million. The odds of winning are about 1:175 Million. This means that if Rocky fills out 175 million lottery tickets, he is guaranteed to make a profit. (Assuming that he doesn’t have to share the prize.)

http://www.nylottery.org/ny/nyStore/cgi-bin/ProdSubEV_Cat_403_SubCat_337550_NavRoot_320.htm
http://www.megamillions.com/

But Rocky doesn’t want to stand at his local Seven-Eleven and fill out 175 Million tickets. (After he enters  Trophy Wife’s birthdate, the dog’s birthdate,  and his lucky number from inside of a Chinese Fortune Cookie, he won’t remember what numbers to pick.)  So instead, he will ask the Seven-Eleven lottery clerk for  175 Million “Quick Pick” tickets.

A “Quick-Pick” is a computer-generated random number lottery entry. The computer picks the numbers, so Rocky doesn’t have to think that hard.

Alas, this won’t work either. Because even if Rocky buys 175 million Quick-Pick, there is some chance that he will receive duplicate Quick-Pick entries … and there is some chance that he won’t receive the winning combination.

The chance of getting a duplicate Quick-Pick should be the same as the chance of winning the lottery. But in Rocky’s case, achieving this result is an illustration of really bad luck.

So Rocky poses the following math question: What is the OPTIMAL number of Quick-Pick tickets to buy? (The optimal number should maximize the chance of getting the winning combination, and minimize the chance of getting a duplicate combination.)

As always, the reader with the best submission will receive a unique prize of dubious monetary value.

Fair coin tosses aren’t fair

To get a fair and balanced perspective on short-term market direction, Rocky occasionally  tosses  a “lucky penny.”   Heads, he buys. Tails, he sells.

Rocky carefully tracks his coin-based trading. The shiny penny with Honest Abe’s countenance (that sits heads-side-up on Rocky’s desk,)  performs better than the new quarters with attractive state images (which sit tails-up on his desk.)  

An intriguing academic paper explains this — finding that Rocky’s coin toss is biased towards bullishness!

“We analyze the natural process of flipping a coin which is caught in the hand.  We prove that vigorously-flipped coins are biased to come up the same way they started.  For natural flips, the chance of coming up as started is about 51%.” 

The paper is entitled “Dynamical Bias in the Coin Toss,”  from statistics professors Diaconis and Holmes at Stanford University, and the full paper can be viewed here :

 http://www-stat.stanford.edu/~susan/papers/headswithJ.pdf

Rocky’s 51% bullish bias  is consistent with the long-term upward drift of prices, but there are more sinister implications for football fans.

In the most recent professional football season, the winner of the overtime coin toss won more than 70% of the games, and since 2002, the coin toss winner is more than 60% victorious. This means that there is the possibility for manipulation, and ceteris paribus, a gambler should always bet on the team that “calls” the outcome of the toss. 

Conclusion: A 1% bias is huge.  In coin-tosses and in life, things ain’t fair!

[Disclosure: Rocky just tossed his penny, and found that crude oil will rise today. But his quarter says that crude oil will decline today. Accordingly, he took the day off.]

Cutlery, children and swans

A child’s bloodcurdling scream always triggers panic in a parent. Especially when sharp cutlery is involved.

The scream came after midnight. Rocky ran into his youngest daughter’s bedroom and saw the knife, and the look of terror on the girl’s face. But he saw no blood on the carpet. His pulse slowed a bit. He quickly examined her fingers, and saw there were still ten. His pulse slowed a bit more.

The child had been trying to activate an Itunes card. The process involves scraping off an adhesive strip to reveal a 16 digit alphanumeric code, and then entering the code into the computer. Unfortunately, her heavy-handed scraping with the knife rendered several of the letters illegible. In the best of times, it is difficult to discern the “I” from the “1,” and the “S” from the “5.” Now it seemed hopeless.

Rocky would sooner cut off his thumb than forfeit an Itunes card . (Particularly since Rocky owns no Apple stock.) He would not go down without a fight.

Rocky brought up his old cobweb-ridden microscope from a dark corner of the basement. Examining the Itunes card under 20x magnification, he narrowed the choices to about 1 in 456,976. After a decent day in the markets, Rocky felt lucky and he chose several probable letters. They typed the code into the computer. Voila! It worked on the first try.

As they high-fived each other, Rocky hoped this experience would teach his daughter at least two life lessons:
1) The role of Black Swans in life.
2) That Rocky is a great Dad.

Rocky pondered the best way to highlight these life lessons.

“Will you take a look at the blog, RockyHumbert.Com?” Rocky asked his daugher. “Just once? You’ve never looked…”

“Dad,” she replied, “Rocky is NOT real. He’s a figment of your imagination.”

“The odds were 1 in 456,976 that this project would succeed,” Rocky replied. “How real is that? Hmmm?”

“Ok, Dad. I guess you earned it. I’ll take a look.”