Sample space for a coin flip

Got it! This site "creationpie.com" uses cookies. You consent to this by clicking on "Got it!" or by continuing to use this website. Note: This appears on each machine/browser from which this site is accessed.

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

"collectively exhaustive" (contain all possibilities) and be
"mutually exclusive" (no overlap of outcomes).

What is the "sample space" (possible outcomes) for a coin flip?

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

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

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

What if the coin rolls off the table?
What if the coin disappears?
What if something else happens?

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".

In Python, the bottom value is None.
In SQL, NULL is the bottom value
In JavaScript, undefined is the bottom value

Related fields:
Statistics
Informatics
Computer Science

What is the (approximate) probability that the flip of a fair coin results in heads?

If you say it is one half, then that is the probability to you. I can see the coin and to me it is either zero or one - I know the result.

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.

▶

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 obvious outcomes are "heads" and "tails".
Before the flip, the outcome is not known, represented by "bottom" or "⊥".
Some events are so rare, such as "edge", that in case of that event, it is done over.
Some events may be ignored, such as losing the coin, as it is done over.

The probability of a flip of a coin requires (information) knowledge of the point of view.

A coin is flipped.

What are the possible outcomes?
What is the probability of each outcome?

We will ignore the possibility of the coin landing on edge. The "bottom" value "⊥" is the state where the result is not yet known.

A fair coin is a coin for which the probability of heads and tails is each 0.5.
A biased coin is a coin for which the probability of heads and tails are not each 0.5.

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

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.

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.

More: John von Neumann

Suppose you have a biased coin that has

Prob(heads) = 0.4, and
Prob(tails) = 0.6.

Here is the method.

Two flips will be done.

How many possibilities are there?

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?

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

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?

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?

What is the chance that in 10 coin flips, you will get every coin flip as specified? For example:

H H H H H H H H H H (all heads)
T T T T T T T T T T (all tails)
H T H T H T H T H T (some other pattern)
... and so on ...

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

In general, a probability of 1/10^3*m is about the same as a probability of 1/2^10*m. That is, 10*m coin flips. When m is 1 then 10 coin flips.

So the following hold as quick approximations.

To convert 10^m to binary, multiply m by 10 and divide by 3 to get 2^10*m/3.
To convert 2ⁿ to decimal, multiply n by 3 and divide by 10 to get 2^3*n/10.

So the following are quick approximations.

10¹⁵ ≈ 2⁵⁰
10¹⁸ ≈ 2⁶⁰
10²¹ ≈ 2⁷⁰
10²⁴ ≈ 2⁸⁰
10²⁷ ≈ 2⁹⁰

What is your probability of winning the super state 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 = 1/10^3*3 which is about 1/2^3*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.

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

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.

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

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

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

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?

Are these the only possible results?

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

Before 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. Bottom

During 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.

▶

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.