Sunday, February 01, 2015

Odds are Good

This is annoying, because it (again) shows that people have an impoverished sense of what probability means.

Because they can't wrap their head around an "event."

The incident:  Michigan lottery "picks" same 4-digit number two days in a row?  What are the odds of THAT happening?  It must be (pick one:  God.  Fraud.  Sign that probability isn't real.  Etc.)

Well, it depends on what you mean by "THAT" in the paragraph above.  Consider:
  • 44 US states have lotteries. Let's say they all have a 4-digit game, to keep things simple.
  • There is a lottery result every day, all 365 days per year.
  • The chances of hitting any given number is 1/10,000  (because 0000 is a possibility, up to 9999, by ones)
 So, if "THAT" is the chance of the Michigan lottery having exactly the same number two days in a row, then THAT is pretty unlikely.  It's 1/10,000 every day, because it's the chance of hitting yesterday's number again today.

But if we are talking about one state lottery somewhere (there was nothing special about it being Michigan, ex ante) on some day in given year (there was nothing special about those two days), then THAT is just the chance one lottery out of 44 picks the same number on consecutive days, out of 365 (since it could happen on the first day, but that would be across years).

If there are 44 4-digit lotteries every day, and the probability of getting a different number in each particular lottery is 9,999/10,000, that means that the probability of duplicate numbers in SOME state (out of 44), on a given day, is .00439.

But we do that 365 times per year.  Since the chance of no duplicates in all 44 states, on a given day, is .9956, the chances of no duplicates for a year is .9956^365 or .2007.

If that's right (and I'm just doing this back-of-the-envelope, so I've probably made a mistake in logic or calculation!), that means that in any given year the chances of a duplicate lottery, two consecutive days the same number, in some state, is about 80%.

Does that sound right?  If you carry out to multiple years, say 5 years, the chances of getting at least one duplicate in at least one state are better than .999.  It will be a little more complicated in real lotteries, because they are not all simple "pick four digits between 0 and 9," but the same sort of logic applies.

With the caveat, again, that I have likely made a mistake.  The question, then, is whether consecutive duplicates are really as common as this calculation implies.  Thoughts?

Example.....  Example.....  Example..... Explanation.

Excellent example...

Lagniappe:  Scott de M suggests an exercise, left to the reader:  Prove that some athlete, somewhere, in some sport, has a jersey number that matches both  his age and number of wins he has played in.

7 comments:

Thomas W said...

People certainly don't understand probability in many cases.

The other side is a comment in Nicholas Talib's "The Black Swan" -- For the classic "if you flip a fair coin 999 times and it comes up heads each time, what's the chance it will come up heads again"? The conventional response is 50%. Mr Talib says the answer should be "heads", because at that point the probability that this isn't a "fair" coin is higher than the probability the coin will come up tails.

Tim Worstall said...

The probability is 1.

As it usually is of events that have happened.

:-)

Gene Callahan said...

'Mr Talib says the answer should be "heads",'

"Heads" is a probability?!

Michael said...

This reminds me of the birthday problem teachers like to pull out when covering probability in high school.

P becomes greater than 0.5 that two people in a class will share a birthday when the class size is larger than something like twenty-three (forget exactly; too lazy to look it up or redo the calc.)

always amazes kids that something that seems so unlikely actually isn't

Pelsmin said...

It's not just the unwashed/uneducated who get it wrong. So much of probability runs contrary to intuition that the PhD's screw up too.
I remember 25 years ago when Marilyn vos Savant, a columnist, ran a now famous puzzle asking whether a contestant on a game show should switch the door they picked after they then see one of the other two opened to reveal no prize. (see it at http://marilynvossavant.com/game-show-problem/) Her answer was that you doubled your odds of winning if you switched from one unknown door to the other.

She received hundreds of letters from PhD professors telling her she was a bonehead. The odds were 50/50! It made no difference if you switched!
They signed their names to angry letters asking her to apologize.

Not being as smart as those guys, I ran a small monte-carlo style simulation on my IBM 286, and saw that she was right. I had no idea why, but her answer was legit. Since then, I've learned enough about Bayesian logic to understand why people believe something when it's not true. If you take a little information but close your mind to additional, relevant information, you may not realize that there's a better answer out there.

Steve in Pennsylvania said...

Hey:
Odds are that some leader, somewhere will hit it big.
Mugabe wins in Zimbabwe.

http://news.bbc.co.uk/2/hi/africa/621895.stm

John Covil said...

Hopefully in phrasing the question, Taleb doesn't specify it's a fair coin. Because a right answer that requires violating a premise of the question is rhetorical dirty pool.

(note: I'm not actually that cranky about it and I get the point he's making)