Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematicianTadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing: If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses.

Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematicianTadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing: If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses.

Really? Why would that be true?

It's one of those counter intuitive logic problems that make no sense at first, but when you do the math it makes every sense. Like the Monty Hall problem or the birthday problem.

The solution here is quite simple;

If Alice fails to get a tail after a head, she only needs to toss again and keep doing it until she gets a tail. Bob on the other hand, if he doesn't get a head after his first head, needs to start over again. Alice's trials don't reset after a failure, whereas Bob's trials do. So on average he will take longer.

Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematicianTadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing: If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses.

Really? Why would that be true?

Actually, the head and tail don't have an equal chance of appearing. The head side is slightly heavier, which is why it tends to land with that side up more often.

Slovenia Member #172924 February 9, 2016 46 Posts Offline

Posted: March 16, 2016, 12:01 pm - IP Logged

Quote: Originally posted by Luminus on March 16, 2016

Actually, the head and tail don't have an equal chance of appearing. The head side is slightly heavier, which is why it tends to land with that side up more often.

That's only an added effect. Even if we had a fair coin whose both sides weighted exactly the same, Alice would still be winning. Even if we said that Bob needs two tails instead of two heads, with a fair coin, Alice would still be winning.

United States Member #51269 April 3, 2007 529 Posts Offline

Posted: March 17, 2016, 12:01 pm - IP Logged

Quote: Originally posted by elios311 on March 16, 2016

That's only an added effect. Even if we had a fair coin whose both sides weighted exactly the same, Alice would still be winning. Even if we said that Bob needs two tails instead of two heads, with a fair coin, Alice would still be winning.

How do you know? Has this been tested? I can't see why that would be, since the head and tail are just engraved on the coin. Technically, both sides are equal, meaning there is no head or tail.

Slovenia Member #172924 February 9, 2016 46 Posts Offline

Posted: March 17, 2016, 4:19 pm - IP Logged

Quote: Originally posted by Luminus on March 17, 2016

How do you know? Has this been tested? I can't see why that would be, since the head and tail are just engraved on the coin. Technically, both sides are equal, meaning there is no head or tail.

Yes, I just wrote an application to test it and am getting 4 for Alice and 6 for Bob on average. I will post it here as soon as my no links restriction is lifted. You can also test it yourself by simply tossing a coin, but it might take a while to get a large enough sample to calculate averages.

But you don't really need to test to know this. As explained earlier, Alice's trials don't reset whereas Bob's do. It has nothing to do with coin's physical characteristics.

United States Member #567 August 14, 2002 484 Posts Offline

Posted: March 18, 2016, 11:48 am - IP Logged

The truth is that the numbers are ONLY used to identify the balls and distinguish the balls from one another. No mathematical formula is going to predict the Powerball lottery game. For example, the balls selected from Wednesday's Powerball drawing were 10, 12, 13, 46, 50 and red ball 21. if you picked 9, 11, 14, 45, 49, and red ball 20, you were no closer to the jackpot that if someone had picked 1,2,3,4,5 and Powerball 10. You either matched those balls or you didn't.

Smart lottery winners form trust to claim their winnings. They send an attorney to the lottery headquarters to claim the prize in trust, so that ONLY the name of the trust is revealed. And they tell NO ONE, especially relatives.

If you ever win a lottery and you are single, the only person you should ever marry is someone who was truly in love with you BEFORE you won the jackpot!

United States Member #51269 April 3, 2007 529 Posts Offline

Posted: March 18, 2016, 12:14 pm - IP Logged

Quote: Originally posted by rundown99 on March 18, 2016

The truth is that the numbers are ONLY used to identify the balls and distinguish the balls from one another. No mathematical formula is going to predict the Powerball lottery game. For example, the balls selected from Wednesday's Powerball drawing were 10, 12, 13, 46, 50 and red ball 21. if you picked 9, 11, 14, 45, 49, and red ball 20, you were no closer to the jackpot that if someone had picked 1,2,3,4,5 and Powerball 10. You either matched those balls or you didn't.

Maybe or maybe not. I read about a mathematician who claims that you can predict lottery numbers more likely to be drawn:

NY United States Member #23835 October 16, 2005 3474 Posts Offline

Posted: March 18, 2016, 6:17 pm - IP Logged

Quote: Originally posted by Luminus on March 17, 2016

How do you know? Has this been tested? I can't see why that would be, since the head and tail are just engraved on the coin. Technically, both sides are equal, meaning there is no head or tail.

Testing will confirm it, but simple logic explains why it has to be true.

Both Bob and Alice need to start by getting the first heads, so that part will be exactly equal. They also both have the same 50% chance of success on the next flip. It's what happens the 50% of the time that they fail on the next flip that results in the difference.

When Alice fails to get a tails she already has the first heads that she needs in order to get a heads followed by a tails. That means she can succeed with one flip after the failure: first H > failing H > successful T

When Bob fails to get his consecutive heads he has a tail, and therefore has to flip at least once more to get the first heads, and then flip again to get a second heads: first H > Failing T > first H > successful H

This is actually somewhat similar to the discovery about prime numbers. It's not the nature of prime numbers or the equal probability of H/T that really matters. It's the sequence from one event to the next.

Slovenia Member #172924 February 9, 2016 46 Posts Offline

Posted: March 19, 2016, 11:42 am - IP Logged

In case anyone wants to simulate this coin toss "paradox" you can download the application from here. Just run the CoinTossParadoxSimulator.exe. If it doesn't work, run the setup.exe first.

You may need to make exclusions in your antivirus software. In case you need to run the setup.exe, if the setup finishes with an error, don't worry It's just an antivirus quirk.

United States Member #158931 September 7, 2014 3725 Posts Offline

Posted: March 20, 2016, 10:43 am - IP Logged

I'm a maths moron, but I think you guys are over-thinking (at least when it comes to the Pick3 games). After reading this thread, I set out to find a fresh example of a simple pattern in the numbers and it only took me two days to find one. I publicly called last night's Pennsylvania P3 result using software that pulls the world's simplest pattern out of the numbers. It looks like this:

All you do is look for number pairings at a certain number of days. It's not hard, just create a dictionary of every number pair within a window of days (I arbitrarily use 30 to 40, we tried up to 1000 days without better results), then sort the dictionary on the first number + the second number + the number of days, then accumulate a list of the pairings that have happened more than once and reconcile it with the numbers which have fallen most recently within the window you chose to begin with.

I know I'm a knuckle-dragging, mouth-breathing maths moron, but you have to admit, it's a little peculiar how easy it was for me to find a fresh hit based on this idiotically-simple idea. I did it prospectively to give you guys a new example to think about. I know you guys are going to quickly say, "Let's see you do it again!" To which I quickly reply, "Okay." Heh.

Seriously guys, I know I'm just deluding myself, but I would sincerely appreciate it if one of you maths folks would at least take a whack at this just to make once-and-for-all sure that I'm wrong.

United States Member #164727 March 12, 2015 2524 Posts Offline

Posted: March 20, 2016, 11:32 am - IP Logged

Quote: Originally posted by factorX on March 20, 2016

I'm a maths moron, but I think you guys are over-thinking (at least when it comes to the Pick3 games). After reading this thread, I set out to find a fresh example of a simple pattern in the numbers and it only took me two days to find one. I publicly called last night's Pennsylvania P3 result using software that pulls the world's simplest pattern out of the numbers. It looks like this:

All you do is look for number pairings at a certain number of days. It's not hard, just create a dictionary of every number pair within a window of days (I arbitrarily use 30 to 40, we tried up to 1000 days without better results), then sort the dictionary on the first number + the second number + the number of days, then accumulate a list of the pairings that have happened more than once and reconcile it with the numbers which have fallen most recently within the window you chose to begin with.

I know I'm a knuckle-dragging, mouth-breathing maths moron, but you have to admit, it's a little peculiar how easy it was for me to find a fresh hit based on this idiotically-simple idea. I did it prospectively to give you guys a new example to think about. I know you guys are going to quickly say, "Let's see you do it again!" To which I quickly reply, "Okay." Heh.

Seriously guys, I know I'm just deluding myself, but I would sincerely appreciate it if one of you maths folks would at least take a whack at this just to make once-and-for-all sure that I'm wrong.

Wow, you got 070 straight yesterday with just two sets. That's impressive.

Can you do a step by step for Florida so I can understand exactly what you are doing? Thanks.