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Sample space for a coin flip
by RS  admin@creationpie.com : 1024 x 640


1. Sample space for a coin flip
Heads or Tails Heads or Tails or else

To determine the probability that an event happens, one needs to know the sample space.

The sample space consists of all the possible outcomes of an event. The sample space should be What is the "sample space" (possible outcomes) for a coin flip?

2. Result of flipping a coin
Coin flips: heads

What are the possible results of flipping a coin? Here is one result.

3. Two results
Coin flips: tails Heads or Tails

Here are two results of flipping a coin. Are these the only possible results?

4. Three results
Coin flips: side Heads or Tails or else

Can a coin land in the edge? Try it with a nickel. Are these the only possible results?

5. Four results
Coin flips: bottom

The value "bottom" or "" is used to indicate these other possibilities.

In many programming languages, the empty string is "". However, the Null value can be used to indicate "bottom" or "no value".

6. Flip of a fair coin
Heads or Tails Related fields:
 Statistics
 Informatics
 Computer Science

 
What is the (approximate) probability that the flip of a fair coin results in heads?
The field of statistics has a deep connection to computer/information science though it may not be immediately obvious.

Saying: "statistics means nothing to a rock". Unlike data (every particle in the universe), coded information requires an intelligent "point of view".

The entire field of statistics is the concept of known and unknown information by an observer and determining a "best guess" at what the state of the actual information of interest.


7. Review: Sample space for a coin flip
 ▶ 
 + 
 - 
 1 Heads 
 2 Tails 
 3 Edge 
 4 Bottom 

To determine the probability that an event happens, one needs to know the sample space - all possible outcomes. The probability of a flip of a coin requires (information) knowledge of the point of view.
A coin is flipped. Coin flips: bottomWe will ignore the possibility of the coin landing on edge. The "bottom" value "" is the state where the result is not yet known.

8. Fair and biased coins
There is no issue with deciding an issue using a coin flip if the coin is fair.

But what if the coin is not fair

9. Expected value: biased coin flips
Suppose that two individuals want to use a coin flip to settle a dispute, but neither person trusts the other.

How would you use a (possibly biased) coin to perform the coin flip so that neither individual has an advantage?

John Von Neumann provided in interesting solution to this problem.

10. Coin flips
A clever solution to the problem of how to decide coin flips in the presence of possibly biased coins is due to John von Neumann (mathematician, computer scientist, etc.) , famous mathematician, statistician, operations researcher, computer scientist (traditional microprocessors are called Von Neumann machines), etc.

Information sign More: John von Neumann

11. Biased coin
Suppose you have a biased coin that has Here is the method.

12. Method
Tails-tailsTwo flips will be done.

How many possibilities are there?

13. Possibilities
There are 2*2 = 4 possibilities.
HH = Heads Heads (TP = True Positive) HT = Heads Tails (FP = False Positive) TH = Tails Heads (FN = False Negative) TT = Tails Tails (TN = True Negative)

What is the EV (Expected Value) of each of the four cases?

14. Expected value
Here is the EV of each possibility.
EV(HH) = 0.4*0.4 = 0.16 EV(HT) = 0.4*0.6 = 0.24 EV(TH) = 0.6*0.4 = 0.24 EV(TT ) = 0.6*0.6 = 0.36


15. Decision
The two people decide who will have Heads-Tails and who will have Tails-Heads.

Flip twice until the flips come up either Heads-Tails or Tails-Heads. If the two flips come up Heads-Heads or Tails-Tails, the flips are done again until a winner is decided.

Is this a fair way to do it?

16. Expected values
Here are the expected values of the choices.
EV(HT) = 0.4*0.6 = 0.24 EV(TH) = 0.6*0.4 = 0.24 EV(neither) = 0.6*0.6 = 0.52

Since the probability of HT and the probability of TH are the same, this is a fair way to do it.

How many flip pairs are needed, on average, until a winner is determined?

17. Probability estimation
What is the chance that in 10 coin flips, you will get every coin flip as specified? For example: The chance of getting 10 coin flips exactly as specified is 1/1024. This is about 1/1000, or 1/103.
1 / 210 = 1 / 2*2*2*2*2*2*2*2*2*2 = 1 / 1024 1 / 103 = 1 / 1000


18. General rule
In general, a probability of 1/103*m is about the same as a probability of 1/210*m. That is, 10*m coin flips. When m is 1 then 10 coin flips.

So the following hold as quick approximations. So the following are quick approximations.

19. Probability
What is your probability of winning the super state lottery?

20. Super lottery
Your probability of winning the super state lottery is, say, about 1/1,000,000,000, one in a billion (i.e., one chance in thousand million).

What does this mean? Your probability of winning the super state lottery is about 1/1,000,000,000 = 1/109 = 1/103*3 which is about 1/23*10 = 1/230.

Your probability of winning the super state lottery is about the same as flipping a coin 30 times and getting the desired result on each flip.

21. Comparison
To put powers of ten into perspective, the concept of flipping a coin can be used to determine one unit of the measured quantity.
Powers    Coin  Measured of ten    flips quantity --------- ----- -------- 1.00*109     30  Winning the super state lottery (1,000,000,000) 3.16*107     25  Seconds in a year (60*60*24*365.25) 3.65*1012    42  Days in 10 billion years (365.25*10,000,000,000) 3.15*1017    58  Seconds in 10 billion years 1*10*1080   266  Small particles in the known universe 3.15*1097   324  Second for every particle for 10 billion years


22. Estimates
For a quick approximate conversion of a base 10 power to a base 2 power, take the power of ten, divide by 3 (i.e., 3 powers of ten, or a thousand), and multiply by 10 (i.e., 10 powers of 2, or just over a thousand).
1,000,000,000 (billion) 109 2(9/3)*10 = 230

So, 232 is about 4 billion.

23. Twenty questions
Many people have played the game of 20 questions.

In 20 well-chosen questions, you can pick one thing from 220 1,000,000 things (i.e., 20 flips).
1,000,000 = 106 220


24. Practical limit
A practical working limit is much less than 1000 bits of information.

In general, even 200 bits of information would be highly unlikely.

25. Gender change party
What is a "gender change" party?

Is it the same as a "gender reveal" party?

What are the possible results of the gender of an unborn baby (still in the womb) at such a party?

26. Gender reveal
MaleAre these the only possible results?

27. Gender reveal
FemaleI call this a "gender change" party. Why might I do that? (as someone with a extensive background in applied programming language theory)

28. Gender change party
BottomBefore the party, except for the parents, the gender is "unknown" (to me) or "bottom", depicted as "".

In programming language theory, involving mathematical lattices and complete partial orders, the bottom value represents incomplete or missing information. BottomDuring the party, the gender goes from "unknown" or "bottom" to "known" (to me) as either "boy" or "girl".

Thus, in this sense, what is called a "gender reveal" party can, from in information perspective, be considered a "gender change" party.

29. Review: Gender change party
 ▶ 
 + 
 - 
 1 Male 
 2 Female 
 3 Bottom 

In society, a "gender reveal" party is where, unknown to most of those attending, the gender of an unborn baby (still in the womb) is revealed during the party.

At that time, the gender of the unborn baby goes from "unknown" or "bottom" to "known" (to me) as either "boy" or "girl".

Thus, what is called a "gender reveal" party can be considered, from in information point of view, a "gender change" party.

30. End of page

by RS  admin@creationpie.com : 1024 x 640