Coin Toss Probability theory

A coin toss is a single event, uneffected by the previous toss and not affecting the following. Since each toss is independent how do we conclude a thousand or a million tosses will have an increasing convergence on a 50/50 outcome. Isn't it a contracdiction?

My question comes from a discussion here

The poster NSB suggests that house edge has no place in casino managements projections of win loss or other calculations since it only applies at infinity.

I understand what this poster is trying to say and yet accept that results do in real life converge with a greater sample size. But how can one express what seems like a contracdiction in a couple of sentences.

If each toss is an independent trial there is no law that says results will converge until infinity. Since infinity by definition will never arrive and a million heads is possible what is the value of probability in real life.

Is it a case of how the question is expressed? I would welcome an answer that could explain clearly and in simple terms how independent events can seemingly be linked together for any projected result beyond infinity.

Again for the record I accept they do, it just seems that the poster on the above forum seems to have posted the question which does seem a paradox.

Quote: 123

If each toss is an independent trial there is no law that says results will converge until infinity. Since infinity by definition will never arrive and a million heads is possible what is the value of probability in real life.

I beg to differ. The Law of Large Numbers (in particular the weak law) says that as the sample size gets larger the sample average will approach expectations. So you don't need an infinite sample size. In plain simple English it says that the bigger the sample size, the more trustworthy the average will be.

The person who posted that would probably argue there is no such thing as probability, which is something usually said only by those who don't understand it.

Quote: Wizard

In plain simple English it says that the bigger the sample size, the more trustworthy the average will be.

and as the sample size grows the degree of accuracy increases.

From the book "Grinstead and Snell's Introduction to Probability"
found here
Page 4 example 1.3 states:
"The previous simulation shows that it is
important
to know how many trials we should simulate in order to expect a certain
degree of accuracy in our approximation. We shall see later that in these types of
experiments, a rough rule of thumb is that, at least 95% of the time, the error does
not exceed the reciprocal of the square root of the number of trials."

enjoy

The writer is confusing two different concepts of probability. One defines probability as the limit of the relative frequency of the occurance of an event as the number of trials approaches infinity. A key word here is approach. It does not mean at infinity. It suggests, but does not guarantee, that the relative frequency calculated from 1 million trials will be closer to the true, but unknown, probability that is calculated from a smaller number of trials. This is why statistics has methods such as confidence intervals to estimate unknown parameters.

A second concept is based on a sample space, the set of points that represent the outcomes of an experiment. For example, a single coin flip the sample space is{H,T}, for two tosses {HH,HT,TH,TT} etc. The probabilities of a sample point are assigned by logic. The principle of insufficient reason would assign equal probabilities to each sample point. This principle states that if there is no reason to do otherwise, they should be assigned as equal. Note that if you have a two headed coin the principle would not apply. You would assign probabilities in a different manner. Thus for the sample space {HH,HT,TH,TT} the probabilities are {1/4,1/4,1/4,1/4}

In an experiment, the probability of an event is the simple sum of the probabilities of those sample points which are associated with the occurance of the event. In two trials P(H,T)=1/4+1/4=1/2. This is where the 50/50 chance comes from. Note that here there is no mention of convergence or a limit.

And if the number of trials is less than infinity, which it is always, there is no reason why the two definitions of probability will have the same numerical value.

>A coin toss is a single event, unaffected by the previous toss and not affecting the following.
True.

Since each toss is independent how do we conclude a thousand or a million tosses will have an increasing convergence on a 50/50 outcome. Isn't it a contracdiction?

>The poster NSB suggests ...
Oops. Beware. Right at that point you should be aware that you are dealing with a poster who very much likes to pontificate about minutiae and waste time while ignoring the obvious since it is an inconvenient truth.

>that house edge has no place in casino managements projections of win loss or other calculations
>since it only applies at infinity.
I thought you just got finished saying that each and every single event obeyed a certain rule and that at "some large number" the effects of this rule would be made manifest to you. The rule applies to the events. The events are random.
NSB is saying that if you let go of the teapot you do not have to worry about the hot tea splashing onto your feet since the event will not take place until the teapot reaches the floor and shatters, therefore he concludes that the law of gravity does not have meaning until such time as you start screaming in pain from your scalded feet. The very first time that YOU perceive the coin toss, you say "no convergence because its not yet a large number" but the guy who got to the coin tossing event two weeks ago sees that very same coin toss and to him, its now a large number. To the coin, it doesn't even get bored with all those tosses.

>But how can one express what seems like a contradiction in a couple of sentences.
This is no contradiction.

>If each toss is an independent trial there is no law that says results will converge until infinity.
You might want to bear in mind that the casino makes its bank deposits each month, it does not wait until infinity to deposit its profits. And at each and every such bank deposit, those little old guys with the green eye-shades admit that it is not yet exactly 50:50 on the results of the coin toss. Yet each and every month, the casino deposits money in the bank and the players leave town on foot. Infinity has not arrived, but the bank deposit is still made.

Oh, and just to be a pit picky myself,,, a coin toss is NOT a 50:50 expectation. It is a 51:49 expectation biased toward whatever side is "up" at the beginning of the toss.

>In addition, the coin could land on edge.
Coins land on their edge ONLY in university seminars. Have you ever seen it happen in real life?

Quote: FleaStiff

Coins land on their edge ONLY in university seminars. Have you ever seen it happen in real life?

Not from a coin toss. But give me a well-balanced coin, any size, and I can roll it on any flat surface on edge every time.

Actually this has more to do with physics than probability. A tossed coin is spinning and falling, therefore it carries significant kinetic energy. Since coins are unstable on edge but stable on their sides, they tend to fall on one side or the other.

Rolling a coin is the exact opposite. In this case the spin makes the coin act like a gyroscope (the same effect you see on bicycle wheels), imparting stability to the on-edge position. In fact, find a well-balanced coin, one that stays on edge when you set it to on a flat surface, roll it slow enough and it may stop on edge and stay there.

Quote: FleaStiff

>In addition, the coin could land on edge.
Coins land on their edge ONLY in university seminars. Have you ever seen it happen in real life?

According to http://www.codingthewheel.com/archives/the-coin-flip-a-fundamentally-unfair-proposition, a coin lands on its edge 1 in 6,000 throws. I've never seen it happen, but my sample size is much less than 6,000.

Quote: FleaStiff

Oh, and just to be a pit (SIC) picky myself,,, a coin toss is NOT a 50:50 expectation. It is a 51:49 expectation biased toward whatever side is "up" at the beginning of the toss.

The same page as linked above explains this. For it to be true you have to catch the coin. Letting it hit the floor increases the randomness.

I remember a rumor back in the early 70's in northern Nevada that some casino was going to have a coin toss game.
another one about a 50/50 bet with dice with only 2 numbers on a normal die.

Sure glad that never happened!

Unless Wizard you can come up with a new casino game. You would be even more famous.

Wasn't there such a game in the movie Vegas Vacation?

Quote: Wizard

Wasn't there such a game in the movie Vegas Vacation?

I only remember the end of the movie where Sid Caesar dies after winning the Keno ticket.
Good movie to watch this weekend!

It was in a small casino with nothing but silly novelty games, also including rock paper scissors. Chevy Chase lost at all of them.

Two-Up is an Australian casino game based around coin tossing.

Quote: Wizard

It was in a small casino with nothing but silly novelty games, also including rock paper scissors. Chevy Chase lost at all of them.

The setting for that scene was the old Hacienda Casino on Southern Las Vegas Blvd by the Welcome to Las Vegas sign(casino is torn down now).

I'm still thinking about the 51-49 thing for flipping a coin, and I'm still not sure I buy it. The page I linked to above explains it as follows:

A good way of thinking about this is by looking at the ratio of odd numbers to even numbers when you start counting from 1.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
No matter how long you count, you'll find that at any given point, one of two things will be true:

•You've touched more odd numbers than even numbers
•You've touched an equal amount of odd numbers and even numbers

What will never happen, is this:

•You've touched more even numbers than odd numbers.

However, I would assume a legitimate flip would flip at least once. Using the odd and even example, we shouldn't count 1, making tails more likely (starting with heads). However, if we say a legitimate flip would flip at least twice, then we wouldn't count 2 either. I'm not sure where one should start counting. Suffice it to say I'm not convinced the odd/even analogy is the reason for the 51-49 probabilities, and I'm not convinced it is 51-49 for that matter.