If I am tossing a coin...

[flexy]

Someone claims that when tossing a coin

Sequences

1) THHTTHTTHT
2) HHHHHHHHHH

that both sequences have the same likelihood to appear. I say, how can that be. The first sequence would represent a "natural" distribution of H/T since it is more likely that, say, +/- 50% of the sequence comprises of head (or tail). Wouldn't 1) be more likely?

 

[SSSnail]

No, because of 50 - 50.

 

[Zeze]

wiki gambler's fallacy.

 

[lxskllr]

I know little about statistics, but I'd say the specific sequences are equally likely to appear. A random sequence of any order has a greater chance of appearing than all heads.

 

[coldmeat]

nvm

 

[twinrider1]

They're independent events. It doesn't know if the previous flip landed heads or tails.

 

[Crono]

Someone claims that when tossing a coin

Sequences

1) THHTTHTTHT
2) HHHHHHHHHH

that both sequences have the same likelihood to appear. I say, how can that be. The first sequence would represent a "natural" distribution of H/T since it is more likely that, say, +/- 50% of the sequence comprises of head (or tail). Wouldn't 1) be more likely?

They are the same and statistically independent. Don't think of it as heads or tails, think of them as random letters. Would

KEFQERQ or EAFDFQR

be more likely with a random keyboard mashing?
The only reason why you would think HHHHHHHH would be more likely because you perceive it as having order, and order seems unlikely, but you have to quash that kind of thinking because the outcome of one flip does not affect the next in the sequence.

 

[Zeze]

Ask yourself this- do you think the coin gives a crap or has a conscience to care what happened in the past? No. It's always 50/50.

 

[flexy]

But isn't it exactly BECAUSE of the 50/50 "law" that ultimately a relatively even distributed number of Hs or Ts would be more likely?

I understand very well that each throw is independent and that the roulette, dice or coin "has no memory"...but then I could say...oh wait..there is a 50/50 chance with each flip. Therefore, if I flip 20 times, chances are that H MAY appear, say 8 - 12 times..and a sequence of 20 Hs in a row is statistically less likely?

 

[homebrew2ny]

Ask yourself this- do you think the coin gives a crap or has a conscience to care what happened in the past? No. It's always 50/50.

Not to split hairs, but actually it's not...

The heads side of a coin (on most coins including the quarter and nickel) weighs slightly more than the tails side thus creating a small advantage for the heads side to land statistically more. The split is more like 52/48.

 

[Dirigible]

A relatively even distribution is more likely than HHHHHHH. But that's not just a relatively even distribution in the OP. It is an exact specific sequence. Just like HHHHHHHH.

 

[lxskllr]

Therefore, if I flip 20 times, chances are that H MAY appear, say 8 - 12 times..and a sequence of 20 Hs in a row is statistically less likely?

You're comparing a specific sequence to everything else. All heads to every other combination. A more accurate comparison would be the likelihood of exactly alternating heads and tails to all tails. If you were to bet on either happening, you'd probably lose.

 

[EliteRetard]

I happen to know a guy who can flip a quarter to heads ~95% of the time.
LOTS of practice. You flip it just right and you know how many turns it will take before you catch it and voila you get what you want (most of the time).

 

[TuxDave]

Why don't you start flipping coins and let us know which pattern comes up first. Sometimes you just got to try it first and then try to understand it second.

 

[Crono]

But isn't it exactly BECAUSE of the 50/50 "law" that ultimately a relatively even distributed number of Hs or Ts would be more likely?

I understand very well that each throw is independent and that the roulette, dice or coin "has no memory"...but then I could say...oh wait..there is a 50/50 chance with each flip. Therefore, if I flip 20 times, chances are that H MAY appear, say 8 - 12 times..and a sequence of 20 Hs in a row is statistically less likely?

I understand why you aren't getting it because it's a bit counter intuitive to the way our brains normally work since we are designed to look for order and patterns, but think of it this way:

There is nothing special about HHHHHHH versus any other single sequence. It could just as easily come up HHHHHHT or THHHTHH. At first glance you might be inclined to think so because such a "streak" is favorable or unfavorable depending on how you bet, but randomness or chance cares not for whether HHHHHHH... is significant to you, a human. Each individual flip is equally likely (or close enough at 52/48 or whatever it is) to give you heads or tails.

The "big picture" looks special to you because your brain processes it that way. It's one of the psychological aspects of gambling that becomes easy to take advantage of: people buy in to the notion of hot streaks or being "on a roll" and that somehow you are more or less likely to win the next independent bet. Aside from cheating or gaming the system, one roll of a die or flip of a coin does not affect the next. Each individual roll or flip has a set statistical probability (1 in 6, or 1 in 2, respectively) of landing on your choice.

You could get any one of these when flipping a coin 5 times in a row:

HTHTH
HHHTT
HHHTH
TTTTT
THHHH
HTTHT
HHTHT
...
...
...
(you get the idea)

It just looks special when you get TTTTT because that seems special to the normal human being who processes order. When we see order we naturally assume design or luck - either someone got really "lucky" (which in practical, logical terms simply means they found the outcomes to be in their favor after the fact) or was cheating. But again, there is nothing truly special about HHHHHHHHHHHHH or TTTTTTTTTTT given all the other outcomes, not unless you find there is something unfair causing all heads or tails.

 

[SlitheryDee]

Equally likely that it would land on those sides in the specific orders you listed. Not equally likely that one series would be a random distribution that roughly worked out to 50/50 while the other was either one of those exact series you listed.

 

[DaveSimmons]

To keep it simple, for 3 flips there are 8 possible sequences of H, T -- it's 2^3.

Think of them as numbers 1 - 8. Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8 is equally likely to happen.

The chance of HHH is 1 in 8. The chance of HTH is 1 in 8. The chance of TTT is 1 in 8.

For 8 flips, there are 256 different sequences. The chance of HHHHHHHH is 1 in 256. The chance of HTHTHTHT is in 1 256. The chance of TTTTTTTT is 1 in 256.

And so on. The chance of HHHHHHHHHHHHHHHHHHHHHHHH is 1 in 2^24, exactly the same as any 1 specific other 24-flip sequence.

Edit: you're probably confusing the difference between the odds of (roughly as many H's and T's) and the specific sequence THTHTHTH...

for 4 flips, the sequences HHTT, HTHT, HTTH, THHT, THTH, TTHH all have even Ts and Hs so the odds of one of the set appearing are 6 in 16 vs. 1 in 16 for HHHH. But the odds of a specific sequence like THHT appearing are just as unlikely as HHHH.

 

[Colt45]

Yeah, any specific sequence has the odds of (1/2)^flips.

So THHTTHTTHT is just as likely as HHHHHHHHHH, which is to say unlikely. A long string of heads looks un-random, but is just as likely as any other defined string. It's just easier to describe.

In a long enough sample, law of large numbers kicks in, and the amount of T's and H's will almost certainly be 0.5 each, which means on average every second toss will be a head, and the others tails. But this means nothing on a short string.

and yeah, read gamblers fallacy. The system has no memory, even if we do.

 

[Murloc]

I happen to know a guy who can flip a quarter to heads ~95% of the time.
LOTS of practice. You flip it just right and you know how many turns it will take before you catch it and voila you get what you want (most of the time).

that's why I just throw them across the room.

 

[Puppies04]

Someone claims that when tossing a coin

Sequences

1) THHTTHTTHT
2) HHHHHHHHHH

that both sequences have the same likelihood to appear. I say, how can that be. The first sequence would represent a "natural" distribution of H/T since it is more likely that, say, +/- 50% of the sequence comprises of head (or tail). Wouldn't 1) be more likely?

You sir can come to my casino* with all your money.

*When I open it.

 

[Possessed Freak]

This is why I *love* when casinos display the past history at the roulette wheels. Yeah, suckers all.

 

[DrPizza]

OP, what you are correct about is that in 10 consecutive flips, you're more likely to see some combination of heads and tails. If you made a list of every possible combination (permutation), starting with hhhhhhhhhh, hhhhhhhhht, hhhhhhhhth hhhhhhhhtt, hhhhhhhhthh, hhhhhhhhtht, hhhhhhhtth, hhhhhhhttt, etc., you will see that there are 1024 different ordering. EVERY one of these has an equal probability of occurring (namely, 1 in 1024). So, realize that your sequence, thhtthttht (or was it tththhthht? Or was it thhtththtt?) is just as likely as any of these other 1024 possibilities. Of the 1024 combinations only ONE of them is your specific sequence oh heads and tails. In fact, there are 252 different sequences that contain 5 heads and 5 tails. Each is different from the other, and each has the same likelihood of happening.

 

[CurseTheSky]

Both sequences have exactly the same probability of happening. Your brain makes the second case (all heads) seem much less likely, because it is a pattern. Our brains are naturally designed to look for patterns, so when we see one in an otherwise completely random event, we instinctively try to rationalize it, even if our rationalization makes less sense ("must be a lucky coin," "must have found a way to beat the odds," "wow, that's just weird," etc.)

For example, if you were driving and you noticed that all ten cars in front of you were blue, you'd take a mental note of it, and probably remember the event at least for a time. If the ten cars in front of you were red, black, white, white, blue, silver, red, white, black, and blue, you probably wouldn't even notice.