Question

Here's a fun question. Suppose there is a biased coin, and by that I mean it is more likely to land on one side than the other. We do not know the bias.

Devise a game using the coin with two outcomes, which are both equally likely to occur.

I know the solution, so I will give hints if you'd like one.

Answer 1

Quck note: You are not allowed to statistically find the bias by flipping the coin over and over to approximate its value.

Answer 2

get a slot machine with equally likely odds of winning and losing, put said coin in said slot machine. the likelihood of winning or losing is the same by definition. grandma spends her weekends at the casino.

Answer 3

Alchemista, I am convinced that you found this question in one of those mini-comic books you find at the bottom of cereal boxes

Answer 4

theoretically, there is no such thing as truly random. :p

Answer 5

what do you mean mattfeury ?

Answer 6

Oh yes there is. Just record quantum events.

Answer 7

Quantum random number generator, look it up.

Answer 8

will do. yea, i suppose i should have said in computing specifically.

Answer 9

In computer science in most cases a pseudo random number generator is used. Which generates a sequence of "random like" numbers. The generator is seeded usually using the current time along with some other "entropy data" and yes I know its not perfect.

Answer 10

But there are actually quantum random number generators you can plug into a computer.

Answer 11

And you will get truly random numbers that way.

Answer 12

if your analyzing something on the quantum level and can change the outcome by simply observing the event happening. It would no longer be random due to the fact that you have the control to change the possible outcome of said event.

Answer 13

Yes all you have is the biased coin.

Answer 14

zbay, quantum mechanics is statistical in its very nature. The position of an electron is not known only a cloud of probable locations can be calculated.

Answer 15

why don't you just flip the coin a said number of times and record the data and figure out how many time each result comes up and make your probablity from that?

Answer 16

no

Answer 17

zbay: look at my first note at the top

Answer 18

"Quck note: You are not allowed to statistically find the bias by flipping the coin over and over to approximate its value."

Answer 19

Not only that it would only be an approximation.

Answer 20

You would need an infinite number of flips to definitively find the bias.

Answer 21

true but that could be said with any coin you decided to flip.

Answer 22

You can setup a game without knowing the bias of the coin, such that there are two "50 50" outcomes.

Answer 23

1 flip heads or tails?

Answer 24

The game can involve flipping the coin more than one time. That's my only hint.

Answer 25

A single round must have two outcomes each having a probability of 0.5

Answer 26

not going to lie i'm stumped but my guess is to have one flip yielding the results of either heads or tails

Answer 27

With only one flip, you get the bias of the coin. It can either be heads or tails, and one is more probable than the other.

Answer 28

but you wouldn't have the evidence to support your claim of it being bias with only one round. Does the coin have to be flipped?

Answer 29

One round can involve more than one flip. That was my hint.

Answer 30

Try thinking about the probability space of flipping the coin more than once.

Answer 31

alright then my solution is to pick the coin up put it down on heads and pick it up and put it back down on tails. We get one round where both outcomes are equally likely to occur.

Answer 32

That doesn't work, it would count as two rounds.

Answer 33

Because each event has a probability of 1

Answer 34

thats not probability you are certain here

Answer 35

what about direction of the coin?

Answer 36

perhaps we split it up into two even halves by diameter

Answer 37

That's certainly an interesting idea, but you don't have to resort to tricks like that. You can devise a game with normal flips.

Answer 38

By that I mean normal flips without considering direction.

Answer 39

The hint I gave is that a round of the game can involve multiple flips. I will give one more hint: don't consider infinite flips, this is a game that can be played in real life.

Answer 40

hold the coin and flip it once with out looking at it before it lands you will then have the same odds of it landing heads or tails

Answer 41

It's a biased coin, the chance it will land on heads is not the same as the chance it will land on tails. So how does that make sense?

Answer 42

I don't think it is like that the actual event does not have .5 figure it is pseudo just in your head

Answer 43

if you pick up a coin flip it so it makes one revelation and then lands it will land heads or it will land on tails. so you have to figure out a way to make it flip only one time and land flat it takes the bias out of the coin.

Answer 44

Okay no tricks with physics to change the bias of the coin. You don't need any tricks based direction of the coin, or changing the nature of the coin to setup the game.

Answer 45

This is a mathematical exercise don't think about the physics of it.

Answer 46

ewwww....math games 8(

Answer 47

i have the feeling it's going to be the obvious answer that i have looked over 100 times with out considering that is going to solve this problem

Answer 48

He said we can change the setup of the game and have multiple flips ..

Answer 49

What if i draw two coins from my pocket they have probability of .5 and then decide thru the other coin whether the coin is the one i had bet on

Answer 50

I meant draw 1 out of two coins in my pocket

Answer 51

You can only use the biased coin.

Answer 52

Taking out a normal coin defeats the purpose of this exercise.

Answer 53

Shall I give a huge hint?

Answer 54

No

Answer 55

well my best guess was to control the number of flips to fix the odds

Answer 56

but you can't calculate it

Answer 57

it makes it absolute so you wouldn't have to calculate it

Answer 58

what if i had a setup that is not effected by flipping of biased coin

Answer 59

What do you mean by that?

Answer 60

This is the same idea as just taking out another coin. You need to turn the biased coin into one that isn't biased using a game.

Answer 61

Here's my previous hint once again. Consider the probability space of more than one flip.

Answer 62

what do you mean by probability space?

Answer 63

But how do we decide the number of flips

Answer 64

I gave a big hint before. Its a game you could theoretically play in real life, so it doesn't involve an infinite number of flips. So this game (the solution) must involve a finite number of flips for one round.

Answer 65

Someone needs to optimize the Javascript a bit :P

Answer 66

I don't know about you but my browser is not happy with long threads.

Answer 67

mine too

Answer 68

mine either, but if we can't alter the coin, can't control the number of flips, the only thing that is left is controlling the number of attempts of the experiment. So one round is going to give the closest to 50-50 we can get

Answer 69

Think about what you can do with more than one flip in one round.

Answer 70

Wow, those cereal box questions sure generate a lot of interest

Answer 71

http://openstudy.com/groups/mathematics#/groups/mathematics/updates/4e38e9bc0b8bf47d065f8a72

Answer 72

Now

if i flip it more then once
h h h h
t t t t
htth
httt
thhh

Answer 73

You are getting very close now.

Answer 74

Continue in the other thread.