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Flipping Coins October 19, 2008

Posted by Michael in Science.

Ever flip a coin to settle a dispute, or make a tough decision?  Sure you have.

I’ll bet you thought the the odds of an honest and vigorous coin flip were 50/50.

You fool.  There is a dynamic bias in favor of the side that was originally up.  The odds are more like 51/49.

This is just another useful IB hint, which may help get you through the economic crisis if you make enough bets based on coin flipping.


Persi Diaconis, Susan Holmes, Richard Montgomery

Department of Mathematics, Department of Statistics, Department of Mathematics and Statistics

Sequoia Hall University of California, Stanford University, Stanford University Santa Cruz


[Editors note:  ^ Persi, Susan and Richard actually got paid to flip coins!  They gotta be geniuses.]

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. The amount of bias depends on a single parameter, the angle between the normal to the coin and the angular momentum vector. Measurements of this parameter based on high-speed photography are reported. For natural flips, the chance of coming up as started is about .51.

Contrary to what you might intuitively expect, the bias may be more pronounced if you let the coin land on the floor, rather than catching it in your hand.

The full analysis is here.

I personally found this treatise to be a relatively light and entertaining read.  It’s about what you would expect from math geeks in California.  However, if you do not understand all the formulas, please do not ask me questions, because I really don’t have time to explain such simple material to morons like you.  Please direct your questions to Mrs. Peel.


1. geoff - October 19, 2008

Makes the opening to Waiting for Godot slightly more plausible.

2. Wickedpinto - October 19, 2008

I liked reading some of the “fuzzy math” articles that they had in discover, not articles, just small examples.

Like, lets say you pick a case, out of 26, and One of those 26 has a million dollars, what are the odds that your case has that million dollars?

well, it’s 1/26

Now lets say you wittle away 24 cases, and the million dollars is still available in either your case, or in the case that is left, and you have the chance to exchange.

What are the odds that your case has the 1 million dollars?

It’s still 1/26, just cuz the other 24 is gone, doesn’t mean that at the time you chose your case it isn’t 1/26, so your odds are actually favorable for exchanging YOUR case which is still 1/26 likely to have 1 million dollars for the case that is left over, because the one other case, is the only one valued at a 50/50 chance.

virtually useless, but interesting philosophy of the assignment of value.

3. eddiebear - October 19, 2008

Ugh. And I loved prob-stats classes.

4. Wickedpinto - October 19, 2008

Please direct your questions to Mrs. Peel.

So, uh, Mrs. Peel? Whats your situation?

5. Mrs. Peel - October 19, 2008

WP, you’re confusing the Monty Hall problem with Let’s Make a Deal. The difference in a nutshell is that in the Monty Hall problem, Monty has additional information you don’t, whereas Howie Mandel doesn’t know anything more about your case than you do. Switching does NOT improve your chances in Let’s Make a Deal, but it does in the Monty Hall problem.

And my situation is that I am worn out from dancing all night.

6. Sobek - October 19, 2008

“And my situation is that I am worn out from dancing all night.”

The robot? Please tell me it was the robot.

7. Mrs. Peel - October 19, 2008

no, swing dancing. We learned (among other things) a new line dance, the Shim Sham. It was fun.

8. Wickedpinto - October 19, 2008

Doesn’t swing dancing require effort?

nevermind, I’m not interested in your situation.

9. Wickedpinto - October 20, 2008

The probabilities are still the same Mrs. Peel.

Your case, is still 1/26, the OTHER is still 50/50.

fuzzymath wasn’t about fact, it was about how easily statistics could be manipulated, and how those manipulations, basicialy are wrong.


you have a 1 in 2 chance of flipping a coin that will land heads, and the coin lands head.
You flip the coin again, what are the odds? who cares? it still lands heads.
all heads
still all heads.

so assuming that a coin flip is a 50/50 proposition, but you just flipped the coin 10 times and all of those coin flips came up “heads” what are the odds that the next time you flip the coin will be heads?

10. Wickedpinto - October 20, 2008

It’s 50/50.

the toss is Always 50/50, the results are seperate from the toss.

EVERY toss is 50/50 no matter the result.

11. Mrs Peel - October 20, 2008

WP, of course a (fair) coin always has a 50% chance of either heads or tails. Duh. But you’re wrong. If you’re down to two cases, and one of them contains $1 million, then there’s a 50% chance the million is in either case. You’re confusing the Let’s Make a Deal situation with the Monty Hall problem, in which there is an advantage to switch.

Again, the critical difference is that Monty Hall knows what’s behind each door, whereas Howie Mandel (presumably) doesn’t know anything more about the cases than you. There is never a point in Let’s Make a Deal in which Howie provides additional information about the cases the way Monty Hall does about the goats in the Monty Hall problem. The only additional information you get is based on the cases you open. So if you’re down to your case and one other, then there IS a 50% chance that your case contains the million. There is NO advantage to switching. The only reason they offer the switch is because people who have heard the Monty Hall problem but don’t understand it think that there is an advantage.

The statistical principle at work in Let’s Make a Deal is expected value. The banker is offering you a deal that is less than the expected value of your case. If you were right about the 1/26 chance, then the deals that the banker offers toward the end of the game would be drastically different. He is clearly offering a deal based on an equal probability of each of the non-eliminated values being in your case.

Bottom line: the Let’s Make a Deal scenario is NOT equivalent to the Monty Hall problem. The only way they would be equivalent is if Howie were pointing out cases that don’t contain $1 million, as Monty points out doors that have a goat.

12. Mrs Peel - October 20, 2008

that comment was supposed to be joking, but came across kinda harsh. My apologies.

Ok, think about it this way: let’s say you’re playing Let’s Make a Deal with just three cases instead of 26. You’ve selected a case and have two left on stage. One of them contains $1 million. The probability that your case contains the million is 1/3. In Let’s Make a Deal, you then select one of the cases on stage (which each also have a 1/3 probability of containing the million dollars). If you didn’t pick the million, you now have two cases left, each of which has an equal probability of containing the million.

Now let’s back up and say that you’ve selected a case and have two left on stage. Now, Monty Hall walks on stage. He knows what’s in each case. He points to one of the cases on stage and says, “This case does not contain $1 million,” and it’s eliminated. Now the probability that your case contains the million is 1/3, while the probability that the case remaining on stage contains the million is 1/2*, so there IS an advantage to switching.

Do you see the difference? In Let’s Make a Deal, the ONLY information you get comes from opening the cases, so after you eliminate a (non-million) value, the probability that the million is in any given case is equal for each case. In the Monty Hall problem, Monty provides additional information, so when he eliminates a case, that changes the probabilities. Your case elimination is random in Let’s Make a Deal. Monty’s case elimination is NOT random in the Monty Hall problem.

I find Let’s Make a Deal fascinating in terms of the choices people make. A purely rational actor would probably take the first deal, because that’s money in your pocket. (The deal is lower than expected value, though…) But people are not rational actors. They’ll dismiss a guarantee of, say, $500, for a chance at a million. Everyone has a different point at which they stop and say, “You know, I have a chance at a million, but I’m being offered X, which is a lot of money. I’ll take X.”

It’s also interesting to see how people’s behavior is affected by the last offer from the banker. If the current offer is lower than the previous, they almost always keep going, even if their chances are worse than previously and the offer is still quite large.

*because either you drew the million or you didn’t. If you did, then Monty can eliminate either case, and the remaining case doesn’t contain a million. If you didn’t, then Monty has to eliminate the last case without the million, and the remaining case has it. So there’s a 50% chance the remaining case contains the million, while there’s still a 1/3 chance your case does. Counterintuitive, I know, but I promise it’s true.

13. Mrs Peel - October 20, 2008

The above is partially wrong. In the Monty Hall situation, there’s a 1/3 chance your case contains the million, and a 2/3 chance the remaining case contains the million. So switching is always to your advantage. But the Monty Hall “paradox” does NOT apply to the Deal or No Deal situation (I called it Let’s Make a Deal in error. Incidentally, while discussing the problem with my friend, I called it Let’s Make a Deal and she called it Who Wants to be a Millionaire. Pop culture fail!).

I’ll post an explanation on my blog after dinner.

14. Wickedpinto - October 20, 2008

Don’t apologize, I came in over my head, so I went teachy on an anectdote.

15. Michael - October 20, 2008

But, WP, you are basically right. The “law of averages” is a myth. If you flip a coin 100 times and it comes up heads every single time, the intuitive reaction is that you are overdue for a tail on the next flip. But, the odds on the next flip are still 50/50.

Well, the odds are actually 51/49 due to the dynamic bias of the flip, if anyone here bothered to read my helpful and informative post.

Judging by the comments, I suppose not.


16. Mrs. Peel - October 20, 2008

The law of averages is a myth, eh? Ask any gamer about his d20.

17. Sobek - October 20, 2008

The study was about coins, not polyhedral objects. Obviously we’re going to need more federal funding for additional studies.

18. Michael - October 20, 2008

polyhedral objects

Hmmm. Apparently the croc god has been losing at craps the tossing of cubical polyhedral objects.

19. Sobek - October 20, 2008

Apparently Michael thinks you play craps with d20.

20. Michael - October 20, 2008

I had to google d20, and I’m proud of that.

21. Wickedpinto - October 20, 2008

You had to google d20?


What an idiot! I guess you’ve never heard the word Vorpal either,
What a loser!

…….no wait.

22. skinbad - October 20, 2008

“Vorpal” is in Jabberwocky. That’s my only connection to it.

23. Wickedpinto - October 21, 2008

When I was a D&D dork, I enjoyed “creating,” unique items that were completely not what they should be. I hated DM’ing, but I would map out and write stuff for use, and my buddy (actually the furry guy) liked them so he would run them out for the people who liked playing (I wasn’t big on playing, I liked the story telling part.) and he would call me up, and I would hear one of my friends in the background cursing me.

“Brian just killed a bugbear….”
“yeah? well?”
He would mention one of the names of one of the things I wrote, and I would laugh.

Nibbler was one of my stupid little creations, If you rolled a true twenty, you morphed into a giant rat, while doing insane damage.

I carried that sorta thing with me when I was on muds.

24. Sobek - October 21, 2008

Michael, just for you I’ll call it an icosahedron from now on.

Dude, have you ever read anything Ace writes?

True story: when I was in high school, I walked into a geometry class one morning and saw the teacher had some large polyhedrons on his desk. I picked up one and said, “oh, it’s an eight-sider.” The teacher was amazed that I knew how many sides it had without counting. My guess is, he’s never slain a kobold in his life.

25. wickedpinto - October 22, 2008

How many times do I have to flip a coin before Mrs. Peel responds to my “whats your situation” question in a way that will make me and the wintersetts happy?

26. BrewFan - October 22, 2008

How many times do I have to flip a coin before Mrs. Peel responds to my “whats your situation” question in a way that will make me and the wintersetts happy?

I’m not sure, WP, but I’m guessing you might find the next undiscovered prime number.

27. Mrs. Peel - October 22, 2008

I thought you’d lost interest, WP. Didn’t I see you asking Mare about her situation a few times?

Sobek, you can actually get a giant fuzzy d20 to hang on your rearview mirror at ThinkGeek. I kinda want one. Or that baseball-style shirt that says “Paladin.” Or…well, about half of ThinkGeek’s inventory…

28. Wickedpinto - October 25, 2008

I’m just a guy who’s interested in situations.

I like euclid, big deal.

29. Wickedpinto - October 25, 2008

Random thing, if Rich ever offers to flip a coin, I suggest you do not touch it.

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