## Welcome to BrainDen.com - Brain Teasers Forum

Welcome to BrainDen.com - Brain Teasers Forum. Like most online communities you must register to post in our community, but don't worry this is a simple free process. To be a part of BrainDen Forums you may create a new account or sign in if you already have an account. As a member you could start new topics, reply to others, subscribe to topics/forums to get automatic updates, get your own profile and make new friends. Of course, you can also enjoy our collection of amazing optical illusions and cool math games. If you like our site, you may support us by simply clicking Google "+1" or Facebook "Like" buttons at the top. If you have a website, we would appreciate a little link to BrainDen. Thanks and enjoy the Den :-) |

# Infinite Flips

### #11

Posted 09 December 2013 - 08:46 PM

Once bias exists, infinity won't guarantee a return to equality.

The probability of equality decreases with bias, but it's never zero unless one of the outcomes never occurs at all.

Note that saying p(H=T) < 1 doesn't prohibit H=T, it only says H=T is not a certain eventual outcome.

*Vidi vici veni.*

### #12

Posted 09 December 2013 - 09:40 PM

Just a side note: it is often falsely assumed that 100% probability is equivalent with absolute certainty. This is one of those cases; even if the coin is perfectly fair, it's **possible** that you will never reach H=T even if you keep flipping forever. The probability is 0, but it's possible. Any given infinite sequence is possible, for example flipping heads every time.

If we look at it backwards, suppose you did flip the coin infinitely many times and recorded your results. Whatever sequence you got, the probability of that particular sequence was 0, but it did happen.

Suppose you generate a random real number between 0 and 10. The probability of generating pi is 0, but it is within the realm of possibilities. It's totally impossible to generate -pi.

In general, suppose we have an event A. If A is absolutely certain to happen, then P(A)=1. But the converse is not true. If P(A)=1, we can't conclude that A is absolutely certain to happen. There is a definition in probability theory, that if P(A)=1, then A happens **almost surely**.

### #13

Posted 10 December 2013 - 01:54 AM

well, i almost surely better understand now. it's actually a little easier for my wee brain to comprehend. the learning experience here is always worth a little humility imho. thanks you guys. oh, one last (not serious) question - what if infinity is odd?

### #14

Posted 10 December 2013 - 04:46 AM

well, i almost surely better understand now. it's actually a little easier for my wee brain to comprehend. the learning experience here is always worth a little humility imho. thanks you guys. oh, one last (not serious) question - what if infinity is odd?

Nice segue to a cute poem from the late Martin Gardner:

**Pi** goes on and on and on ...

And **e** is just as cursed.

I wonder: Which is larger

When their digits are reversed?

*Vidi vici veni.*

### #15

Posted 10 December 2013 - 04:54 AM

*Vidi vici veni.*

### #16

Posted 11 December 2013 - 06:40 PM Best Answer

*Vidi vici veni.*

### #17

Posted 12 December 2013 - 12:20 AM

Spoiler for Suggestive proof

Woah.

### #18

Posted 12 December 2013 - 04:16 AM

### #19

Posted 13 December 2013 - 12:52 PM

How do you partition the flips of a coin with p(tails)=0.4 and p(heads)=0.6 in the scenario where it comes up with ten consecutive tails?

T T T T H H H H - H H is a result that is

**representative**of p(H) = 0.6.

That does not mean that in

**every**ten flips it will happen.

There can be (arbitrarily long) strings of T, so long as p(T) is not zero.

So T T T T T T T T T T

**can**happen.

But p(H) = 0.6 says that,

**on average**, whenever there are 10-T strings there will also be 10-H strings, and they will occur in the following proportions:

T T T T T T T T T T

T T T T T T T T T T

T T T T T T T T T T

T T T T T T T T T T

H H H H H H H H H H

H H H H H H H H H H

H H H H H H H H H H

H H H H H H H H H H

-------------------

H H H H H H H H H H

H H H H H H H H H H

and the hyphens indicate how to partition them.

*Vidi vici veni.*

### #20

Posted 13 December 2013 - 10:20 PM

If P(H)=0.6, the probability of a 10-H string is 0.6^{10} ~ 0.006, whereas the probability of a 10-T string is 0.4^{10} ~ 0.0001. So 10-H strings should be much more frequent than 10-T strings.

#### 0 user(s) are reading this topic

0 members, 0 guests, 0 anonymous users