Has Anyone Ever Flipped Heads 76 Times in a Row?
blogs.scientificamerican.com
blogs.scientificamerican.com
Let's assume instead that it's 10 heads, 10 tails, 10 heads, 10 tails, 10 heads, 10 tails, 10 heads, and 6 tails. The probability of that coming up is exactly the same as 76 heads.
Now, what at the chances of any given permutation coming up? Still the same as 76 heads.
So the author would, presumably, on balance of probability, be convinced that no combination of any 76 coin tosses has ever come up. If 76 heads is unlikely then any other permutation is equally unlikely. As someone who has flipped a coin more than 76 times I can say for certain is wrong. At least 1 permutation has definitely happened.
It is entirely illogical to be convinced that it wasn't heads all the way.
(But, for the record, it wasn't.)
The odds on a given permutation arising are very low.
Edit to address the confusion directly: There's no extra reason to think it wasn't heads all the way, since it's exactly the same probability as any other sequence of 76. And at the same time, we can be pretty sure that no one has flipped most of the possible sequences! There are just so many, it would take forever to explore them all.
- Richard P. Feynman Six Easy Pieces: Essentials of Physics By Its Most Brilliant Teacher
If you enumerate all the possible combinations then of course one of them has come up.
It basically comes down to saying something like this:
1. 1/x ≈ 0 for very large x
2. x0 = 0 for all x
3. So x1/x ≈ 0 for very large x
4. But x/x = 1
Therefore
1 ≈ 0
This reasoning is correct, but the error tucked away in that last "≈" is exactly 1.
So the author would, presumably, on balance of probability,
be convinced that no combination of any 76 coin
tosses has ever come up.
The illogical point only arises if we try to take many of these independent statements all at once. If we take some significant chunk of sequences and state they've each all together never been seen before then we eventually must stop feeling that statement is probable.
One way to clarify the whole thing is to assign a bet to any given probabilistic statement. The author bets $10k that nobody has ever seen 76 heads. I can bet $10k that nobody has ever seen 76 tails. In each case our expected value on these bets is identical. Someone else could bet 10c on each possible permutation and be guaranteed to win, but obviously forced to distribute his bet.
It's not at all illogical for me to feel very confident that your coin flipping sequence wasn't all tails. It's highly illogical for me to believe your coin sequence wasn't in the set "all possible coin flipping sequences of length 76".
How much would you like to bet that this hasn't happened either?
Cards work similarly. Every properly shuffled deck of cards is likely a unique permutation that has never before been seen across the entire history of people shuffling cards.
With 256 bits in the sample space, you it takes an unimaginable amount of effort to find any collisions (unless the hash has a weakness).
Only if you presume that the author would make an error in basic reasoning for you to point out. Being convinced that it's highly unlikely that any particular combination of heads and tails 76 bits long has ever occurred with a fair coin != being convinced that no one has ever flipped a coin 76 times in a row.
I am absolutely sure that if you asked the author "Do you ever think that somebody has had a run of coin flipping that consisted of, in this order, "10 heads, 10 tails, 10 heads, 10 tails, 10 heads, 10 tails, 10 heads, and 6 tails," you would have gotten exactly the same answer.
Do you think that your described run is more likely to have occurred because it looks more representative? Or that since people have flipped coins more than 76 times in a row and had some result that all particular results can't still be improbable to have ever occurred or ever to occur? I don't understand your objection if it's anything other than post hoc reasoning (a particular arrangement happened, therefore its likelihood to occur must have been significant.)
Nope, the author would be convinced that any specified configuration of 76 tosses named in advance would not come up when you spun/tossed your coin 76 times.
Any good?
We rely on this all the time in computer science. What are the chances of guessing the private key of a bitcoin address? Is it possible? Not really, but yes it's technically possible.
So far I've "only" gotten 16 heads in a row after a few thousand flips
best = 0
i = 0
while true
i += 1
count = 0
face = 1
while face == 1
face = rand(0:1)
count += face
end
if count > best
best = count
println("new record: $best in iteration $i")
end
endBut was only 26 times in a row, and wasn't coins. Still, is not impossible, the rest of the article matters. And that it comes from literature remembers me of the case of Teela Brown, author's hand are the ultimate superpower.
[10] pry(main)> choices, tries, current, current_max = [:heads, :tails], 0, 0, 0
[10] pry(main)> loop do
[10] pry(main)* tries += 1
[10] pry(main)* flip = choices.sample
[10] pry(main)* if flip == :heads
[10] pry(main)* current += 1
[10] pry(main)* else
[10] pry(main)* current = 0
[10] pry(main)* end
[10] pry(main)* if current > current_max
[10] pry(main)* current_max = current
[10] pry(main)* puts "New max: #{current_max}, after #{tries} tries"
[10] pry(main)* end
[10] pry(main)* end
So has anyone ever flipped heads 76 times in a row on a fair coin? No. Because we would judge that coin to be unfair.
In reality, an individual's the flipping method is likely to be more deterministic than random and show bias - it's not the coin that is fair or not, but the method of flipping it.
EDIT: I've also noticed that if I choose 'Suppose Heads comes up 50% of the time.' it says 'You'll need an average of 3.00 flips to get a fair coin.' If heads comes up 50% of the time then I have a fair coin, so shouldn't that be 1.00 flip?
If your coin happens to be absolutely fair, it will require on average 3 flips to generate a fair flip via this method. The point is you don't know a priori whether a given coin is fair.
But that's very unlikely to happen if you've got a fair coin. Almost certainly you'll flip ungodly zillions of failing sequences first before hitting any particular pre-ordained target. So with a fair coin the story is that you peep in on a probability lab and then wait there until just before the heat death of the universe and suddenly observe 76 heads.
So honestly, in any fair coin situation your Bayesian probabilistic update going on will actually be predicated on very different evidence.
A Bayesian who sees 76 heads in a row adjusted her priors early on and stopped being surprised after awhile. 76 heads and a tail? That would be weird. 76 heads strongly suggests you were wrong about there being a tail side to that coin.
The frequentist concludes that she's dealing with something anomalous: the distribution is all off. The difference is in how this is built into the math: frequentist statistics can reveal, say, the difference between flip 4 being tails (and all the others heads) and flip 72 being tails (with all the others heads), whereas in Bayesian terms it just 'falls out' of the math.
So here's a fun one. Given the 10h-10t-10h-10t-10h-10t-10h-6t: which model does the best job of predicting (correctly imho) that we're more likely to see four more tails than anything else?
So the question probably ought to be "what are the odds of 76 heads and it being detected correctly + the odds of not getting 76 heads but believing you did". In the real world, the last part would obviously dominate and so we are more likely to see false positives than real results.
enum Side { Heads, Tails }
void Main() { Func<Side> generator = () => { return Side.Heads; };
int heads = 0;
while (heads < 76) { if (generator() == Side.Heads) ++heads; else heads = 0; }
Console.WriteLine("Got {0} heads in row.", heads); }
Neither of those things are unlikely. People probably also exist who accidentally shot themselves while cleaning a gun on April 22, 1981, or who slipped in the shower and died within an hour of telling someone "I could slip in the shower and die."
There's no quantitative way to assert whether or not someone at some point in human history flipped a coin that landed heads 76 times in a row.