Home > General, Markets > Fair coin tosses aren’t fair

Fair coin tosses aren’t fair

August 6, 2009

pennyTo 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 :


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.]

  1. masteroftheuniverse
    August 6, 2009 at 12:00 pm

    I love coin tosses, and whenever someone says that the toss of a coin is 50/50, I offer them 2:1 that in ten tosses that they won’t get 5 heads and 5 tales. You’d be surprised how many people bite at that one.

    You should see what I do with a full shot glass and a stack of dimes:)


  2. August 6, 2009 at 12:11 pm


    First they tell me life’s not fair and now coins. Where am I going to find a good source of randomness?

  3. August 6, 2009 at 3:00 pm

    For truly random numbers, Rocky recommends http://www.random.org … which allegedly generates random numbers from “atmospheric noise.” It’s unclear whether “atmospheric noise” means they are intercepting twitters between consenting adults, or a purely physical phenomenon.

    Additionally, because Rocky is “statistically challenged” he not only gets to park in those handicapped spaces, but also doesn’t know whether climate change impacts Random.org’s “atmospheric noise” algorithm. As a mathematician and physicist, perhaps you can answer this question?

    Lastly, Random.Org lists testimonials from random satisfied customers. For example: “Thank you for random.org. When I’m not running massive monte carlo simulations, my favorite use of your site is deciding where my work colleagues and I will go to lunch every day. We tried voting, but we were unhappy with the results — someone would end up disappointed that their choice wasn’t picked. So, now we put all of our choices into the “list randomizer” and go whereever your random choice picks. Oddly, we’ve been going to McDonald’s every day for the past 8 years.”

  4. August 6, 2009 at 3:21 pm

    Jeff writes: ” You should see what I do with a full shot glass and a stack of dimes:)”

    Rocky would have guessed, “make a bunch of crank calls from the payphone in the bar,” however, the price of a call went up from a dime to a quarter some years ago. Come to think of it, the bar doesn’t even have a payphone anymore!

  5. masteroftheuniverse
    August 6, 2009 at 10:12 pm

    Back in the old days when they had the rotary pay phones, I used to carry a wire with a needle soldered to the end. I would ground one end of the wire to the ground of the phone booth and put the needle through a hole of the mike on the handset and would get a dial tone. There was no need for using money. When I wanted to make long distance, I had a cassette of the sound of quarters dropping through a phone and when the operator would tell me to deposit money, I’d play the tape. Back then, the operator listened to the tone made by the coins. I did that stuff in junior high until I built a Blue Box and then never paid for a phone call again.

    With the shot glass and dimes, if one fills the shot glass up with water, one would be surprised at how many dimes you can slide into the glass without water spilling over. The surface tension of the water allows for an amazing amount of dimes to fit in the glass. Make sure the glass is clean and the water is pure and you can have a field day.

  6. ld
    August 7, 2009 at 7:34 am

    Rocky – You may want to try the Eagle. It doesn’t matter whether you end up with heads or tails as long as you get to keep the coin.

  7. August 14, 2009 at 7:14 pm

    Ah, statistics. Its value and relationship to reality is well-illustrated by that three-door problem thingie.

  8. masteroftheuniverse
    August 14, 2009 at 7:30 pm

    @fakename, I have always preferred studying probabilities over statistics as probabilities can look forward while statistics are looking in the rear view window.

  9. August 14, 2009 at 8:02 pm

    Um…the three-door problem is really about probabilities. But as a mathmatically challenged person, I still have to ask the question: aren’t probabilities based on past performance? Which sorta brings us back to statistics?

  10. masteroftheuniverse
    August 14, 2009 at 8:18 pm

    No, probabilities aren’t based on past performance. Not even in the races, despite what some pundits would like you to believe.
    If probabilities were based on past performance, and a coin landed on heads 20 times in a row(assuming the coin wasn’t gaffed), would the past performance affect the probability on the next toss? I think not, but there are plenty of gambling systems out there that make that assumption, and none of them are net winners. If a roulette number pays 35 to 1, and there’s 37 numbers(1-35 and 0, 00) on the wheel, past performance doesn’t matter a bit. I’d pay with a smile every time I lost if I took a bet on a roulette number considering the huge overlay on the game. Same thing with any other casino game. It’s always good to pay short odds and that’s what keeps the lights on in Vegas. Anyways, isn’t the Monte Hall Paradox(actually Bertrand’s Box Paradox) a decision that has nothing to do with past performance?

  11. August 14, 2009 at 9:08 pm

    Rocky guesses that he and fakename were both sleeping in the back of their P&S class … and are paying the price for those ZZZZ’s to this day. Nonethess, Rocky guesses that there’s a bit of imprecise semantics (but correct “memory”) in fakename’s comment that “probabilities …based on past performance.” He’s probably referring to conditional probabilities — (i.e. Baye’s Theorem) … and how future probabilities change as information is revealed….which is the Monte Hall Problem which he colloquially refers to as “the three door problem thingie.”

    Nonetheless, the most troubling problem is why three adults are blogging on a friday night about Baye’s Theorem. Don’t they have anything better to do ? Or are they really just playing internet poker on the other computer screen????


  12. August 15, 2009 at 7:55 am

    I’m glad you jumped back in, Rocky, because it was clear that I was in over my head 🙂 Fakename is a “she”, and blond at that, which may explain things better. As for the blogging on Friday night, I’ve made a Note to Self not to do that any more, so as to give the illusion of having a scintillating social life.

  13. August 15, 2009 at 8:47 am

    Fakename: Rocky won’t make any derogatory comments about your recently disclosed gender after Lawrence Summer’s experience at Harvard…. See:


    Here’s a riddle: why is a man with blond hair spelled without an “e,” but a blonde lady is spelled with an “e”. And the pale FURNITURE color is spelled “blond” too? Perhaps the answer is in that last sentence!

  14. August 15, 2009 at 2:02 pm

    Fakename is familiar with Summers’ comments and believes he is now in a more appropriate position, since cash is gender-neutral. You can’t have female ones, and male fives.
    Now as for the “e”, you are stepping into Fakename’s territory, language and spelling. “Blond” can be used for either sex. But “blonde” is female. Apparently furniture is male. Who knew?

  15. August 15, 2009 at 6:32 pm

    Fakename: Yes, furniture is male. But “sometimes a hatstand is just a hatstand.”

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