Toss a coin 10 times.

[snico]

one run is more likely than the other, thats a mathematical fact.

No, that's not true.

Coin no. 10 has no knowledge of what the previous 9 coils fell, so coin 10 has a 50/50 chance of coming up heads, just as coins 1, 2, 3 etc do.
HTHTHTHTHT is exactly as likely as HHHHHHHHHH or HHHHHTTTTT

But we are talking about a sequence or run of events. Yes each coin is 50/50.....but what is the probability, thats how we work it out.

Flip 10 coins...HHHHHHHHHH is 1/1024
Flip 10 coins...HTHTHTHTHT is 252/1024
Flip 10 coins...HHHHHTTTTT is 252/1024

Flip 1 coin...H is 1/2

So to flip a coin 10 times, with a result of all heads, as a sequence not a single event, is very unlikely.

Each of those three sequences are specific sequences so all thre are 1/1024.

The chances of getting 5 heads in a run of 10, there position within that sequences being unimportant, THAT is more likely than any of the sequences mentioned above, but once you state what the position is, then the odds are 1/1024.

That is true yes, I was just choosing random HT combinations, any combination containing 5 heads and 5 tails will be 252/1024, and so on.

 

[chapeau]

Right so now that's been done to death how about a harder one.

I toss a coin ten times and repeat this 100 times. How many times out of the 100 series would I expect to get 5 heads and 5 tails?

You only get the glory if you show your workings

 

[UriBentMySpoon]

I was the original op and no that wasn't the question, the question was about the sequences given. If I had said chances of 10 heads against 5 heads and 5 tails in any sequence then yes, 10 heads extrememly unlikely.

 

[snico]

Right so now that's been done to death how about a harder one.

I toss a coin ten times and repeat this 100 times. How many times out of the 100 series would I expect to get 5 heads and 5 tails?

You only get the glory if you show your workings

Well if you get 252 combinations of 5 heads and 5 tails from tossing the coin just 10 times. You could possibly get 5H and 5T 25200 times. Doesn't mean you would expect that though, but its possible.

 

[jeds]

If you toss a coin ten times there are 1024 possible unique outcomes. Because each sequence is unique (in other words can only occur once) each has a 1/1024 possibility of occuring. So it's correct to say that any sequence is equally likely as any other. But that is only relevant when comparing the likelyhood of one unique sequence occuring rather than another unique sequence. It is not the same question as asking how likely is it that you will toss ten heads.

Example; What is the likelyhood of tossing one head in ten? There are 10 possible sequences containing one head. therefore the answer is 10/1024. (less than 1%)

2 heads = 45 possible sequences = 45/1024 (about 4 and a bit %)
3 heads = 120 possible sequences = 120/1024 (nearly 12%)
4 heads = 210 possible sequences = 210/1024 (shade over 20%)
5 heads = 252 possible sequences = 252/1024 (nearly 25%)

And no heads is 1/1024 which is not even close to 1%.

So you can see that each unique sequence is (mathematically) equally likely but the probablility of getting 5 heads (and therefore 5 tails) in any sequence is much greater than one head or no heads etc.

 

[chapeau]

Well done chaps! Most people think it's 50% and they'd be wrong.

It's binomial theorem and not for the faint of heart.

Of course I'm stupid so wouldn't understand.

 

[jeds]

Right so now that's been done to death how about a harder one.

I toss a coin ten times and repeat this 100 times. How many times out of the 100 series would I expect to get 5 heads and 5 tails?

You only get the glory if you show your workings

Toss a coin ten times you'd expect to get 50/50 heads and tails. So that's five of each out of ten. x100 = 100.

sorry. naught blind.

 

[chapeau]

Er, think you got it right the first time...... About 25 of the 100 trials would result in 5 heads and 5 tails.

That binomial stuff....

 

[jeds]

It's 25% of the trials x2 - because each sequence can happen backwards as well.

I'm confusing my unique sequences now. It's 25%. But you'd expect to get 5 heads and 5 tails every time.

 

[chapeau]

Well I did say I was stupid or was that somebody else?

 

[snico]

Er, think you got it right the first time...... About 25 of the 100 trials would result in 5 heads and 5 tails.

That binomial stuff....

How did you work that out?

Flip coin 10 times......x100?

 

[chapeau]

Jeds did the working above.

Binomial theorem is to do with the fact there are multiple paths to the solution so you essentially add them up. When you do that then there are more results that are not 5H/5T than are 5H/5T so the probability is set accordingly.

You can use it to see whether a coin is loaded. 6H/4T isn't for example as that happens enough by chance.

If you have a large enough sample you start to get repeatable and predictable results.

 

[snico]

Jeds did the working above.

Binomial theorem is to do with the fact there are multiple paths to the solution so you essentially add them up. When you do that then there are more results that are not 5H/5T than are 5H/5T so the probability is set accordingly.

You can use it to see whether a coin is loaded. 6H/4T isn't for example as that happens enough by chance.

If you have a large enough sample you start to get repeatable and predictable results.

I understand binomial theoreom, pascals triangle etc.

I'm looking again at the possible combinations, my fault.

 

[wreckedit]

When a well known illusionist tossed a coin 10 times on camera it came up heads every time. They didn't show the other 7000 odd times it took to get that result though.

 

[UriBentMySpoon]

When a well known illusionist tossed a coin 10 times on camera it came up heads every time. They didn't show the other 7000 odd times it took to get that result though.

Derren Brown, in the end credits of that programme he showed attempt after attempt, (not all of them obviously) think it took him all day to get the one result he wanted.