Little puzzles

[cultureofdoubt]

One of the newsletters I get regularly has little counter-intuitive logic problems:

First one I give just out of interest:
Poll Coin tossing

Second one has a more interesting twist I thought of:
Poll False sentencesNow, the meta-twist to this one that occurred to me is that none of them have any mistakes - as they're written exactly as intended in order for the puzzle to work.... so

Comments

Re: ***brain exploding***

[cymruangel]

I quite agree - my poor head won't even let me contemplate those.

You've taken me all the way back to the "supplementary" maths classes we had at school for a year (aka torture by logic puzzles).

Darn you sciencey/mathsy types and your twisty makes-mathematical-sencse-but-just-sounds-wrong logic!

[typical]

If you keep tossing a coin, for exactly half the tosses the probability you will have tossed an equal number of heads and tails is zero :p

[cultureofdoubt]

Answer to the first one is that the probability decreases (ignoring typical's odd/even thing) - your chance of being close to 50% rises, but your chance of being at exactly 50% decreases.
Second one is traditionally that the first one is true, the second is a paradox (and therefore not true, but also not false?) and the third one is false.

[evath]

I have an issue with the coin tossing my first toss has a probabity of 0 that I will have equal tails and heads, the second will have 1/2. Thats an increase? Am I wrong or is thatit only work after a certain point?

[ext_2366]

It never works for an odd number of tosses.

0 tosses - p == 1
1 toss p == 0
2 tosses p == 1/2
3 tosses p == 0
4 tosses p == 6/16 ? am I right here?

[typical]

Yes, you're right.

Formula is 0.5^n x n! / ((0.5n)!^2)
n == 2, f(n) == 1/4 x 2 / (1^2) == 1/2
n == 4, f(n) == 1/16 x 24 / (2^2) == 6/16
etc.