Suppose I draw one ball after another from the bag without replacing the earlier selected balls. I can ask what is the probability that the first of the two balls I draw is black? Now suppose I look at the first ball and see that the ball is black. What is the chance now I get two black balls? If the first ball had been white, what is the chance that the first two balls are black? The answer is that this cannot happen, so the result must have probability 0.
The way these ideas are treated is as follows: If X is an event we say that the probability of X is P X. So, how might we deal with the probability that event X will occur given that event Y has occurred? And what about the probability that Y will occur given X has occurred? Clearly these are sometimes different! So let us "invent a notation" for such "conditional" probability occurrences.
How can one calculate P X Y? We will make the following "definition. Events A and B are represented by the circles, and outcomes that have properties A and B are shown by the circle's overlap. If we want to know the probability of B occurring given that A has occurred, P B A , we can restrict our attention to things inside A since we know A occurred. Now to find out what chance there is that B occurred given A occurred means we want to see "what part" of A is represented by the overlap of A with B. If A and B are independent, that is, they do not affect each other's occurrence, then we should have that P A B should be equal to P A.
This is the multiplication rule for independent event probabilities. Using the notations developed above, here are the answers to the questions above, as well as some other probabilities, about probabilities of drawing 2 balls one after another from a bag with 3 balls, 2 of which are black.
The "birthday surprise" is but one of many situations in which people's intuition for the nature of probability has to be "trained. Martin Gardner had a playful streak and often had columns which had whimsical goals. In his Scientific American column of April he showed a plane map which purportedly required 5 colors to color the faces. The rule is that if two regions share an edge they should get different colors. The face-coloring problem for maps asks: What is the minimum number of colors with which one can color the regions of a map drawn in the plane so that regions faces which share an edge receive different colors?
This problem, which has a long history, resulted in one of the most famous "false proofs" in the history of mathematics. Alfred Kempe, a lawyer by profession but with a significant record of accomplishments as a mathematician, claimed to have proved what came to be called the 4-color conjecture.
This conjecture asserted that any plane map could be colored with 4 or fewer colors. Kempe's proof had a subtle error and that error was pointed out by Heawood, who showed that one could prove that plane maps could always be 5-colored. The proof of Kenneth Appel and Wolfgang Haken, that any planar map could be 4-colored, was apparently completed in and did not appear in print until a year later. So Gardner's column occupied the niche of high interest in the status of the conjecture that pre-dated the appearance of a proof.
Some controversy accompanied the proof because it involved the use of a computer in a way that precluded humans being able to follow all of the steps. While many mathematicians were not bothered by this aspect of what Haken and Appel did, there was considerable discussion of the matter by mathematicians and others.
However, this proof, while more streamlined than the earlier one, still involved the use of a computer. So yes, Gardner's column was an April Fools' joke. Here is a coloring of the challenge map above, designed by William McGregor, which was found by Stan Wagon. While all plane maps are 4-colorable, finding a 4-coloring for a particular map is often a challenge. I was in the audience to hear Martin Gardner but I never met him in person. However, in a variety of ways our paths crossed. Gardner first came to my attention when as a teen on a trip to Florida, an automobile accident stranded me and my parents in Winter Haven, Florida, while our car was being repaired.
My "lifesaver" for this period was a local library within walking distance of where we were staying, which was where I got my first introduction to Scientific American magazine and the columns of Martin Gardner. On returning to NY I started a subscription to Scientific American , and my subscription has never lapsed. Eventually, when I had large enough living quarters I started to acquire Gardner's books. HIs books were initially easier to consult than trying to go back to his Scientific American columns.
However, there was also the fact that when his Scientific American columns appeared in book form, Gardner updated information about the mathematics in the columns with additional details about what was new about the problem since it had initially been published.
go site One wonderful aspect of Gardner's Scientific American columns was that he and an "army" of appreciative, motivated and talented people including some who were professional mathematicians thought about the questions that he raised. In some cases his columns dealt with issues which, at the time he wrote about them, were not fully understood.
The Mathemagician and Pied Puzzler: A Collection in Tribute to Martin Gardner [ Elwyn R. Berlekamp, Tom Rodgers] on lapetela.gq *FREE* shipping on. Summary. This volume comprises an imaginative collection of pieces created in tribute to Martin Gardner. Perhaps best known for writing.
So his books allowed him the opportunity to "update" columns with new ideas and results when the columns were collected for reprinting. I also had an indirect "meeting" with Martin Gardner. While I was a graduate student at the University of Wisconsin in Madison, I became interested in what have come to be called Eberhard Type Theorems , in honor of the blind 19th century geometer Victor Eberhard. In conjunction with this I became interested in ways to tile convex polygons with various shapes, in particular with equilateral triangles and squares.
The diagram below shows a "patch" of tiles of a portion of the plane which uses only equilateral triangles and squares. Can you pick out in this tiling corners for convex polygons with 3, 4, 5, and 6 sides? What other values of n are there so that there is a convex polygon with n sides which can be tiled with equilateral triangles and squares all having the same side length?
In thinking about this one can have vertices of the tiling polygons that are not vertices of the convex polygon. For example, above you can find a convex pentagon which is tiled with two squares and a triangle. The answer to this question is a nice application of some simple ideas from elementary geometry so I submitted it as a problem to Mathematics Magazine.
Here is what appeared in November Proposed by Joseph Malkewitch, University of Wisconsin. For what values of k is there a convex polygon with k sides which can be dissected into squares and equilateral triangles which have the same length of side? The solution by Michael Goldberg appeared in May, There are still some open questions related to the shapes of those n -gons that can occur for some values of n. I was very pleased when Martin Gardner picked up on this problem and used it in his book Knotted Doughnuts and the later compendium Colossal Book of Short Puzzles and Problems.
However, Joseph Malkewitch is really me! Vould I lie to you? Gardner fans were disappointed when Gardner no longer continued to write his column for Scientific American. The magazine tried to find a "replacement" for Gardner by having various authors and columns that tried to capture the spirit of Mathematical Games. Eventually, however, Scientific American resigned itself to the fact that Gardner really could not be "replaced! Fortunately, the spirit of what Gardner started with his columns and books continues via a series of events to remember him and his name.
This is done via the conferences and workshops in different locations that have come to be known as Gatherings for Gardner. It is a wonderful tribute to Gardner that what he valued so much will continue to inspire people in the future. Mathematics is a magical subject and you don't have to be a "trained" mathematician to savor, love, and participate in the magical world of mathematics. A list of some of Martin Gardner's books: 1.
New Mathematical Diversions 4. The Numerology of Dr. Matrix 5. The Unexpected Hanging 6. If you write glibly, you fool people. Just as knowing how a magic trick is done spoils its wonder, so let us be grateful that wherever science and reason turn they finally plunge into darkness.
Many professional mathematicians regard their work as a form of play, in the same way professional golfers or basketball stars might. Mathematical magic combines the beauty of mathematical structure with the entertainment value of a trick. In 'Preface', Mathematics, Magic, and Mystery , ix. Mathematics is not only real, but it is the only reality. That is that entire universe is made of matter, obviously.
And matter is made of particles. So the entire universe is made out of particles. Now what are the particles made out of? The only thing you can say about the reality of an electron is to cite its mathematical properties. Also, first sentence as filler, with citation, after Washek F.
Mathematics, a creation of the mind, so accurately fits the outside world. The formulas of physics are compressed descriptions of nature's weird repetitions. Phrenology is the only major pseudoscience I know about that once flourished around the world and has since faded away. Politicians, real-estate agents, used-car salesmen, and advertising copy-writers are expected to stretch facts in self-serving directions, but scientists who falsify their results are regarded by their peers as committing an inexcusable crime.
Yet the sad fact is that the history of science swarms with cases of outright fakery and instances of scientists who unconsciously distorted their work by seeing it through lenses of passionately held beliefs. Excerpted in John Carey ed. Speaking about symmetry, look out our window, and you may see a cardinal attacking its reflection in the window. The cardinal is the only bird we have who often does this. If it has a nest nearby, the cardinal thinks there is another cardinal trying to invade its territory.
It never realizes it is attacking its own reflection. Superstrings are totally lacking in empirical support, yet they offer an elegant theory with great explanatory power. We can then ask the unanswerable question, "Why this set of equations? Thanks to the freedom of our press and the electronic media, the voices of cranks are often louder and clearer than the voices of genuine scientists. The last level of metaphor in the Alice books is this: that life, viewed rationally and without illusion, appears to be a nonsense tale told by an idiot mathematician.
At the heart of things science finds only a mad, never-ending quadrille of Mock Turtle Waves and Gryphon Particles. For a moment the waves and particles dance in grotesque, inconceivably complex patterns capable of reflecting on their own absurdity. The universe is made of particles and fields about which nothing can be said except to describe their mathematical structures. In a sense, the entire universe is made of mathematics. If the particles and fields are not made of mathematical structure, then please tell me what you think they are made of!
There are, and always have been, destructive pseudo-scientific notions linked to race and religion; these are the most widespread and damaging. There is still a difference between something and nothing, but it is purely geometrical and there is nothing behind the geometry. Irving Joshua Matrix. Gathering for Gardner Martin Gardner bibliography.