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Flipping a coin

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313083.  Wed Apr 09, 2008 5:10 am Reply with quote

I'm giving this its own thread, so I can post some things that were decided by the toss of a coin. It'll make good notes if we run with the Q.

Question: I'm about to toss "this"* coin, what is the percentage chance of it landing as a head?
*Camera pans onto Stephen with the coin about to be flipped heads-up.

Forfeit: 50%, 49%, "it may land on its side"

Answer: 51%

More here:

313085.  Wed Apr 09, 2008 5:11 am Reply with quote

In May last year, a Conservative councillor in Lincolnshire held his seat by the toss of a coin.

Christopher Underwood-Frost and LibDem John Birkenshaw both won 781 votes in the West Lindsey district.

We did ask whether we between ourselves could agree to disagree with the toss of a coin... but the law's the law, and the law needs to be changed in my view. it appears that this is a British law.

313089.  Wed Apr 09, 2008 5:14 am Reply with quote

There are a few odd established ways of deciding an election result when the votes are tied after several recounts, aren't there? I've heard of drawing names out of a hat before. I'm sure there was once a reference to a game of poker, though that might have been across the Atlantic.

313091.  Wed Apr 09, 2008 5:16 am Reply with quote

Italy found their way into the final of the 1968 European Championships (in Football) after winning a thrilling coin-toss against the Soviet Union. They had drawn their quarter-final 0-0 and went on to win the tournament.

Liverpool made it to the semi-final of the European Cup in 1965 after beating Cologne by choosing correctly. (altogether more satisfactory that their victory-by-penalty over Arsenal last night),9204,770769,00.html

313093.  Wed Apr 09, 2008 5:17 am Reply with quote

In Isaac Asimov's story of fortuitousness "The Machine that Won the War", one of the men working the machine admitted at the end that he had actually made vital decisions by tossing a coin.

313095.  Wed Apr 09, 2008 5:21 am Reply with quote

Yup Bob, well remembered. This from SQUIRE:

A hand of poker is used to resolve a tie in New Mexico elections. The last time this occurred was in 1999, when Republican Jim Blanq defeated Democrat Lena Milligan in the race to become a local judge after tying on 798 votes.


313102.  Wed Apr 09, 2008 5:27 am Reply with quote

Tossing the coin is the usual way of deciding tied counts in this country; it's surprisingly common.

313109.  Wed Apr 09, 2008 5:31 am Reply with quote

The horserace "The Darby" had its name decided by the toss of a coin, had the Earl of Derby lost, the race would be called "The Banbury" after Henry William Bunbury.

s: The Spectator (London); Mar 22, 2008

313116.  Wed Apr 09, 2008 5:33 am Reply with quote

Bert Brocklesbury from Yorkshire had a lot of soul searching to do as a WWI methodist. In the end, he tossed a coin to see what God wanted him to do and refused all military duties.

The decision cost him dear:

he and his fellow absolutists were held in medieval dungeons beneath Richmond Castle as the Army vainly tried to break their wills. When this failed, Bert and 34 others were shipped to France, court-martialled, and sentenced to death, although the sentences were instantly commuted to 10 years' penal servitude.

The Sunday Telegraph (London); Mar 2, 2008

313353.  Wed Apr 09, 2008 9:57 am Reply with quote

Coin tossing to decide the result of a tied election is required by law in several US states. There's a newspaper article about it here.

313358.  Wed Apr 09, 2008 10:01 am Reply with quote

There is a Wikipedia article about coin flipping which has a few interesting nuggets in it. I do not understand this one, but maybe the mathematically-minded among us might:
Wikipedia wrote:

Coin flipping in telecommunications

There is no fair way to use a coin flip to settle a dispute between two parties over distance for example, two parties on the phone. The flipping party could easily lie about the outcome of the toss. In telecommunications and cryptography, the following algorithm can be used:

1. Party A chooses two large primes, either both congruent to 1, or both congruent to 3, mod 4, called p and q, and produces N = pq; then N is communicated to party B, but p and q are not. It follows N will be congruent to 1 mod 4. The primes should be chosen large enough that factoring of N is not computationally feasible. The exact size will depend on how much time party B is to be given to make the choice in the next step, and on party B's expected resources.
2. Party B calls either "1" or "3", a claim as to the mod 4 status of p and q. For example, if p and q are congruent to 1 mod 4, and B called "3", B loses the toss.
3. Party A produces the primes, making the outcome of the toss obvious; party B can easily multiply them to check that A is being truthful.

The article can be found in full on

313429.  Wed Apr 09, 2008 12:15 pm Reply with quote

It is using what is I think referred to as a 'trapdoor code'.

Being congruent to 1 mod 4 means that the number is of the form 4L+1 for some integer L. Similarly congruent to 3 mod 4 means of the form 4M+3 for some integer M.

It is quite easy to show that if you multiply pairs of numbers (either of the first type or of the second type) the result will always be of the form 4N+1 (e.g. (4A+3) x (4B+3) = 16AB + 12A + 12B +9. This is 4(4AB + 3A + 3B +2) + 1).

If you choose very large numbers for p and q and multiply them together to give the very large number N, then it is very hard to find p and q given only N. (I could explain this more, but for instance if p and q have 63 digits, it would take a quadrillion years or some such for a computer to find p and q!). So if you send B this large number N and give him a few seconds to reply he would have to give a random call 1 or 3. Once A has then sent him p and q it is the matter of a moment (on a computer!) to multiply the two numbers and check that they are indeed the factors of N.

This type of code is used to keep all our internet transactions safe. There is an unsolved mathematical hypothesis (The Riemann Hypothesis) concerning the distribution of prime numbers. Some people think that if this is ever solved, then internet security might be compromised overnight - leading to worldwide meltdown.

313441.  Wed Apr 09, 2008 1:06 pm Reply with quote

I thought I'd solved the Riemann Hypothesis once but it turned out that I had forgotten to carry the 2 in the second line. Duhhh ...

679882.  Sun Mar 07, 2010 12:43 pm Reply with quote

Does anyone have the clip of this as it would give me an interesting end to a mathematics lesson that I will be teaching soon?

679893.  Sun Mar 07, 2010 1:19 pm Reply with quote

Actually, I don't believe the OP question strictly makes any sense because statistics don't apply to single events.

I think he could say, "If I were to toss this coin in this manner a number of times, what would be the expected outcome as a percentage of heads"?


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