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(Math) Probability and Stat: Losing with Heads or Tails

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PostPosted: Thu Jan 05, 2006 2:14 pm    Post subject: (Math) Probability and Stat: Losing with Heads or Tails Reply with quote

Losing with Heads or Tails
From Science News for Kids March 3, 2004.
Emily Sohn

Heads, you win. Tails, you lose.
It turns out that coin tosses may be less fair than you might think. A new mathematical analysis even suggests a way to increase your chances of winning.

People use coin tosses all the time to make decisions and break ties. You've probably done it yourself to decide who gets the last piece of pizza or which team gets the ball first. Heads or tails? It's anybody's guess, but each side is supposed to have an equal chance of winning.

That's not always true, say mathematicians from Stanford University and the University of California, Santa Cruz. For a coin toss to be truly random, they say, you have to flip the coin into the air so that it spins in just the right way.

Most of the time, though, the coin doesn't spin perfectly. It might tip and wobble in the air. Sometimes it doesn't even flip over.

In experiments, the researchers found that it's practically impossible to tell from watching a tossed coin whether it has flipped over. A tossed coin is typically in the air for just half a second, and a wobble can fool the eyes, no matter how carefully you watch.

To see how wobbling affects the outcome, the researchers videotaped actual coin tosses and measured the angle of the coin in the air. They found that a coin has a 51 percent chance of landing on the side it started from. So, if heads is up to start with, there's a slightly bigger chance that a coin will land heads rather than tails.

When it comes down to it, the odds aren't very different from 50-50. In fact, it would take about 10,000 tosses for you to really notice the difference.

Still, when you're gunning for that last piece of candy, it can't hurt to have a leg up, no matter how small.—E. Sohn



Questions to explore further this topic:

What is coin flipping (tossing)?

What are random variables?

What is probability?

What is a percent?

What are permutations and combinations?

What is statistics?

What are graphs?

Interesting applets regarding probabilities

Birthdays (See how frequent people would have the same birthday)

Cards (See how lucky you are in guessing the color)

Collecting cards (See how many draws you need to collect all 52 cards)

Roll the dice (See what combinations you would get from a given number of throws)

Statistics and Probabilities in Chemistry:


Here is an online textbook (advanced) on probabilites:


Last edited by adedios on Sat Jan 27, 2007 3:09 pm; edited 2 times in total
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PostPosted: Wed May 24, 2006 12:41 pm    Post subject: The license plate game Reply with quote

The license plate game Or something to think about while caught in traffic
STAR SCIENCE By Carlene P.C. Pilar-Arceo, Ph.D.
The Philippine STAR 05/25/2006

One activity I entertained myself with as I was growing up was looking at vehicular license plates. Sitting through a traffic jam, I would look at a license plate’s last three digits and, oblivious to my parents’ complaints about the traffic, happily spin numbers in my head. It made me an ideal passenger, always quiet and content.

At that time, plate numbers were still of the "letter-digit, digit-digit-digit" combination. Sometime in my teen years the license plate combinations changed to "letter-letter-letter, digit-digit-digit." Not bothering then with the reasons for the change, I expanded my number game to trying to count the total number of three-letter combinations.

As my study of mathematics deepened (meaning, as I learned how to count properly mathematically), I realized that the change from the five-character combination to the six-character one implied a huge increase in possible vehicle registration, and that traffic jams that would ensue would be just as huge! Now I no longer wonder why traffic jams are inevitable, especially in Metro Manila. This would give my parents, and all commuters, more to complain about.

Let us dissect the present license plate configurations, and concentrate on those from Metro Manila, a.k.a. the National Capital Region (NCR). First, let us look at the first three figures: the three letters. To make computation less complicated, I will fix the first letter, say, N. NCR license plates originally began with N, with the exception of course of government vehicles or "red" plates which start with S, and other special registration plates. (Region I license plates begin with A, Region II license plates with B, Region III license plates with C, and so on.) Next, we note that the letters I, O and Q are never used in license plates. That means that the second and third letters may be any of the 23 remaining letters of the alphabet (letter repetition is allowed). Rules of counting dictate that the total number of three-letter combinations with the above restrictions is 1x23x23 or 529.

Next we look at the last three figures: a three-digit number. This automatically bars 0 as the first digit. However, repetition is allowed so the total number of three-digit numbers is 9x10x10 or 900. (We may avoid the hard math for this by just figuring out how many numbers there are from 100 to 999.)

Again, counting rules dictate that if we want the total number of six-figure combinations with all the above restrictions, that is, three letters where the first one is N and a three-digit number, then the total number of such license plates is 529 x 900 or 476,100.

I am sure you are aware that license plates in the region have gone through several different first letters already, namely, N, P, T, U and W. When I began writing this article new license plates still started with X. Now, I am seeing plates starting with Z! This means that vehicle registration in NCR has used up 6 x 476,100 or 2,856,600 license plates! This translates to 4,560 vehicles per square kilometer of Metro Manila’s 636 square kilometers. As I noted earlier, this still excludes government "red" plates, customized/personalized plates that start with letters other than N, P, S, T, U, W and X, and possibly diplomatic "blue" plates that follow an entirely different configuration.

Fortunately for our health and travel concerns, theory and actual fact differ significantly. Not all of these almost three million vehicles are in use. I expect that quite a number have found their way to different endings – some sitting and rotting away in streets, garages and junkshops, some impounded, some "chop-chopped." Then there is the DOTC directive implemented last year wherein jeepneys and buses aged 15 years and older can no longer have their franchises renewed. Hooray! I hope the implementation has been successful. In any case, the DOTC-LTFRB website says that the actual number of registered motor vehicles in Metro Manila, as of Sept. 30, 2004, was just 852,045 (32 percent of total motor vehicle registration for the entire country). I say "just 852,045" relative to my initial estimate of almost three million vehicles. Then again, this is on the assumption that all vehicles have been properly registered. Still and all, 852,045 is as big as it looks and sounds.

Imagining 852,045 vehicles moving around in a space that is roughly only 636 square kilometers overwhelms me. Metro Manila covers only 0.2 percent of the Philippines’ total land area, yet it is home to almost one-third of all motor vehicles in the country. Sadly, this is not imagination. This is for real. The DOTC-LTFRB figures translate to 1,339 vehicles per square kilometer! Now should we wonder why we have perennial traffic jams and worsening pollution? Why MMDA is continually cooking up schemes such as color-coding? The color-coding scheme aims to reduce vehicular traffic by 20 percent. This 20 percent already means 170,409 vehicles; but a whopping 681,636 remain! Moreover, even with the number of flyovers and fly-unders being constructed, traffic still does not "fly" through these thoroughfares. Though they are welcome attempts at easing traffic flow, most usually turn out to be "crawl-overs" and "crawl-unders." The biggest culprit is sheer vehicle volume. Perhaps the government should also look for schemes of regulating vehicle registration instead of easing traffic flow, which is near impossible if registration is unlimited within a very limited land area.

Check out, if you can, vehicle registration in the other regions. Has Region I used up its A-plates, Region II its B-plates, Region III its C-plates, and so on? If they have not, then lucky are their citizens. While in NCR, by the time we use up our Z-plates,… * * *
The author is an associate professor of Mathematics in UP Diliman. Her research areas include partial differential equations and operations research. A current interest is General Education Mathematics which she has been teaching for the past several semesters.
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PostPosted: Wed Sep 27, 2006 1:33 pm    Post subject: Pure luck Reply with quote

Pure luck
STAR SCIENCE By Carlene P.C. Pilar-Arceo, Ph.D.
The Philippine STAR 09/28/2006

Millions of us line up on Saturday nights, hoping and praying that six carefully chosen numbers from one to 42 will bring us riches galore. The truth is that no matter how much care we take in choosing those six numbers, the chance that all of them (nay, that even just one of them!) will be drawn is very, very small. First of all, I wish to make clear that this article does not intend to discourage you from betting, lest I incur the ire of the PCSO. It merely wants to inform you of the total number of ways of getting a combination of six distinct objects, in any order, from a set of 42 distinct objects. In plain English, we want to answer the question, "In how many ways can we get six numbers from 1 to 42?" More to the point, "What are the chances of my bet(s) being drawn?"

It’s not as daunting as it sounds. Think of the tambiolo (I don’t know what they call the machine that spews out the balls with numbers written on them so kindly allow me to use the well-known term, tambiolo.) Think of what happens to the 42 different balls as one by one, six of them are drawn, and not returned to the tambiolo.

Unang bola. Of course, at the very start, there are obviously 42 different choices inside the tambiolo. After one ball is drawn from/spewed out by the tambiolo, obviously, 41 balls remain.

Ikalawang bola. Now, there are 41 different choices. After another ball is drawn from/spewed out by the tambiolo, obviously, 40 balls remain.

Ikatlong bola. Now, there are 40 different choices. After another ball is drawn from/spewed out by the tambiolo, obviously, 39 balls remain.

This goes on until…

Ikaanim na bola. Now, there are 37 different choices. After another ball is drawn from/spewed out by the tambiolo, obviously, 36 balls remain.

I hope you’re still with me, because the math just starts here. The rules of counting tell us that to get the number of ways six such balls can be drawn, we must first multiply the numbers 42, 41, 40, down to 37. Thus, we get 42x41x40x39x38x37 = 3,776,965,920. Whew, almost four billion possibilities! That calls for a brief pause in the computing. (Meaning, this is not the answer yet.)

To help us catch our breath, let’s go back to the betting game. Although we may not follow any particular order in writing down our six chosen numbers, most of us encircle them (on the betting form), in ascending order (by row, left to right, top to bottom). For example, I bet on the combination 36, 37, 1, 4, 6, 5 and I encircle the numbers in the order 1, 4, 5, 6, 36, 37. Indeed, on the stub returned to me the numbers are written in that order – 1, 4, 5, 6, 36, 37. Does this alter my bet in any way? Not at all! In fact, if these six numbers do come out, maybe they won’t even come out in those two earlier arrangements. Let us say they come out in the order 4, 5, 37, 6, 1, 36. Does this mean I didn’t win? On the contrary, I still do! As long as the six numbers show up, the order in which they come out does not matter. In mathematics we call this a combination, a selection or grouping wherein order is not important.

Since we have mentioned mathematics, let us resume computation. We have just established that the arrangement 36, 37, 1, 4, 5, 6, in this discussion, is no different from 1, 4, 5, 6, 36, 37, from 4, 5, 37, 6, 1, 36, and any other arrangement of the six given numbers. However, in our initial computation (3,776,965,920, remember?) they are all accounted for. Meaning, if there are 100 possible arrangements of 1, 4, 5, 6, 36, 37 (in mathematics these are called permutations), all 100 possibilities should be counted only once; also meaning that among the 3,776,965,920, they have been double-counted 99 times! To do away with this double-count, rules of counting tell us that we should divide 3,776,965,920 by the total number of possible arrangements of the six numbers. By a procedure similar to the tambiolo draw, we get 6x5x4x3x2x1 possible ways for the six numbers to be arranged, a.k.a. six-factorial, or 720 ways. This means there were 719 double-counts! Proceeding with the required division, 3,776,965,920 / 720, we get 5,245,786. This is the answer we set out to find.

So, without further ado…

"In how many ways can we get six numbers, in any order, from 1 to 42?" In 5,245,786 ways.

"What are the chances of my bet(s) being drawn?" If you bet on one combination, one out of 5,245,786, or 0.00002 percent. If you bet on two combinations, two out of 5,245,786 or 0.00004 percent. If you bet on three combinations… If you bet on 2,622,893 combinations, 2,622,893 out of 5,245,786, or 50 percent…If you bet on all 5,245,786 possible combinations, 5,245,786 out of 5,245,786, or 100 percent!

It is inevitably true that the more you bet, the more chances you have of winning. Even if you fail, the PCSO guarantees your hard-earned money will be put to good use so more bets translate to more goodwill. (In that sense you still win.)

Some points to ponder:

Let us say you bet on all 5,245,786 possibilities, are you sure you will get back all the money you bet – more than P50 million – and the money of all the other bettors? Think again and don’t jump for joy yet. What if someone else also bet on the winning combination? Ouch!

An article on the Internet says that your chances of being struck by lighting in a given year may vary between one in 400,000 and one in 240,000, or between 0.00025 percent and 0.00042 percent. (You can boost those figures by going golfing or swimming during a thunderstorm.) That’s up to 21 times higher than your chances of winning this lottery!

Finally, no matter how small those percent figures are, no matter how close to zero they are, it is still valid to say they are not equal to zero. So can you get struck by lightning? Sure, it happens. Can you win the lottery? Sure, with pure luck. * * *
The author is an associate professor of Mathematics in U.P. Diliman. Her research areas include partial differential equations and operations research. A current interest is General Education Mathematics which she has been teaching for the past several semesters. E-mail her at
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PostPosted: Fri Feb 16, 2007 8:12 am    Post subject: Numbers are just numbers, but how you grasp them fills in de Reply with quote

University of Oregon
15 February 2007

Numbers are just numbers, but how you grasp them fills in details

Presentation may be important to help people measure risks

Quickly now, which is a higher risk that you will get a disease: 1 in 100; 1 in 1,000; or 1 in 10? Choosing the correct answer depends on a person's numeracy – the ability to grasp and use math and probability concepts, according to a presentation at the annual meeting of the American Association for the Advancement of Science.

The scenario was part of a series of experiments with University of Oregon students. In this case, 96 percent correctly chose 1 in 10, a 10 percent chance. However, that response came from a more-educated, college-going crowd. The numbers of correct answers fell significantly when put before less educated and older Americans, said Ellen Peters, a UO courtesy professor of psychology and senior research scientist with Decision Research, a non-profit research institute in Eugene, Ore.

"It's interesting that many people can't get simple questions like this, but what I'm really interested in is: Does it matter to decision making, such as choosing medical options, picking stock or mutual funds?" Peters said. "It turns out, yes, it does. In risk communication, you can talk about a 1 percent chance of disease or a 1 chance out of 100. Logically, those two presentations are equivalent, but they elicit different risk perceptions depending on your number ability."

To help people better comprehend numbers and assess certain risks, presentation may be vital, Peters said. The National Adult Literacy Survey, she noted, found that 47 percent of Americans don't have minimal math skills necessary to use numbers imbedded in printed materials.

"It's not that low numerate people are stupid," Peters said. "It's just that high numerate people transform numbers better. A lot of decisions involve numbers. It turns out that how good you are with numbers influences not only whether you understand them, which is how we traditionally think about math abilities, but it influences how we process the information into decisions."

An example, she said, involves two jars of jelly beans. A large jar holds nine colored beans among 100; a smaller jar contains 10 beans with one colored. Choosing a colored bean means you win a prize. If given one chance, from which jar would you refer to draw? Low numerate people, Peters said, are more likely to be drawn by emotional factors and draw from the larger bowl simply because they see more winning beans. High numerate people, however, see the big picture and draw affective meaning from a comparison of probabilities: nine in 100 is 9 percent, whereas 1 in 10 is 10 percent, therefore they have "and feel they have" a better chance going for the smaller jar.

To show the impact of numbers in risk perception, Peters told students that a patient, Mr. Jones, had been evaluated by a respected psychologist for discharge into the community. One group of students learned that of every 100 similar patients, 10 percent would likely commit violence. The second group was told that for every 100 such people, 10 were likely to do so.

High numerate students saw risks as equal, Peters said. "Low numerate people didn't see as much risk for Mr. Jones' potential for violence if told only that there is a 10 percent chance. We found that when low numerate people were told instead that there was a 10-in-100 chance, they could picture 10 people running around going crazy and realized that Mr. Jones may be one of them."

In another experiment, Peters showed two charts, one a commonly used Adjuvant Decision Aid that helps women choose which therapy to accept, if any, after breast-cancer treatment. The other chart was a simplified version. Subjects were asked to study the charts and recommend a therapy.

The Adjuvant Decision Aid features cluttered bar graphs and shows that 70 of every 100 women are alive in 10 years after choosing no additional therapy, that 23 died because of cancer and another seven died of other causes. The chart also shows the impacts of chemotherapy alone, hormonal therapy alone and combined therapy. The simplified chart shows four bars, clearly showing that 70 of 100 women are alive after 10 years with no therapy; 74 of 100 are alive after chemotherapy, 76 are alive after hormonal therapy, and 79 of 100 are alive after combined therapy.

After viewing the standard chart, 44 percent of subjects, regardless of numeracy, chose no additional therapy. Subjects viewing the simplified chart had a different take. Only 4 percent of low numeracy subjects chose no more therapy, while 6 percent with higher math abilities agreed.

"Numeracy and format interact in comprehension," Peters said. "With adjuvant, people with lower numeracy understand much less of the information, but even people high in ability didn't do that great. With the improved format, everybody got better in comprehension almost equally, which means the low numerate subjects were helped the most."

The National Science Foundation, National Institutes of Health and private industry, including Blue Cross Blue Shield Association, fund her research.

Source: Ellen Peters, UO courtesy professor of psychology, 541-485-2400,

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PostPosted: Tue Feb 27, 2007 7:38 am    Post subject: Archaeologists and Clergy Slam Jesus Film Reply with quote

Archaeologists and Clergy Slam Jesus Film

By Marshall Thompson
Associated Press
posted: 26 February 2007
01:19 pm ET

JERUSALEM (AP)—Archaeologists and clergymen in the Holy Land derided claims in a new documentary produced by the Oscar-winning director James Cameron that contradict major Christian tenets. "The Lost Tomb of Christ,'' which the Discovery Channel will run on March 4, argues that 10 ancient ossuaries—small caskets used to store bones—discovered in a suburb of Jerusalem in 1980 may have contained the bones of Jesus and his family, according to a press release issued by the Discovery Channel.

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PostPosted: Sat Dec 22, 2007 6:26 am    Post subject: Questionable Numbers for a Questionable Remedy Reply with quote

Week of Dec. 22, 2007; Vol. 172, No. 25/26

Questionable Numbers for a Questionable Remedy
Echinacea might be useful as a cold remedy or preventative, but science hasn't shown it yet
Julie J. Rehmeyer

When you first feel the sniffles and wonder what to grab from your medicine cabinet, perhaps you should first check some numbers. Especially if one of your choices is echinacea.

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