Physics: Principles and Problems

Chapter 1: A Physics Toolkit

In the News

How many ways can we shuffle a deck of cards? You'd be surprised.

September 2004

Even with computer solitaire?

Playing cards have been manufactured in their present form of 52 cards for centuries, perhaps thousands of years. Millions of decks have been made, and millions of people have shuffled their decks of cards at the start of every card game. Has every possible shuffle been achieved? Since we have the ability to work with very large numbers using scientific notation, we can answer that question.

First, convince yourself that the number of possible ways that three cards could be shuffled is 3×2×1=6. We call this number '3 factorial' and write it as 3!.

Four cards could be shuffled 4×3×2×1=24=4! ways. Thus the number of possible shuffles for a 52-card deck is 52!, which is 8×10⁶⁷ shuffles. Try multiplying the first ten numbers of the series on your calculator.

Extreme assumptions, all for the sake of argument.

Now, consider the most extreme case, which would be that everyone who ever lived on Earth shuffled a deck of cards one time each second since the beginning of human existence. How many shuffles would that be?

The present population of Earth is about 6.2 billion people. Estimating the total number of people who have ever been born is difficult because we do not know exactly how many years modern people have lived on Earth and we do not know exactly how the human population has fluctuated over much of that time. However, several recent models have estimated that roughly 100 billion people have lived on Earth. Finally, fossil records indicate that modern humans appeared roughly 100,000 years ago, so we will assume that people have been capable of shuffling cards for 100,000 years. Thus, the assumptions that we are making for our calculations are that 100 billion people have shuffled a 52-card deck one time each second for the last 100,000 years.

This is where we use our scientific notation skills to work with really big numbers:

The number of seconds in 100,000 years is 1×10⁵ yr × 365 days/yr × 24 hr/day × 60 min/hr × 60 sec/min = 3×10¹² sec. That is how many times one person could have shuffled a deck of cards in a 100,000 years, assuming one second per shuffle.

Finally, 100 billion people could have accomplished 100×10⁹ people × 3×10¹² shuffles/person = 3×10²³ shuffles. Now, let us compare this enormous number with the number of possible shuffles of a deck of cards. That would be 52! possible shuffles minus 3×10²³ shuffles accomplished. Doing the subtraction on our calculator, we get 8×10⁶⁷ - 3×10²³ = 8×10⁶⁷, within the accuracy of the calculator. Thus the number of shuffles that could have been accomplished under the most ridiculous of circumstances is 67-23=44 orders of magnitude below the number of possible shuffles.

So the answer of whether or not every possible shuffle has been achieved is………

Activity:

Reference:

http://www.dartmouth.edu/~chance/course/topics/winning_number.html

http://www.wcsscience.com/deck/ofcards.html

Random numbers, epidemics and winning streaks

October 2004

Do you know what “random” really means? Have a look at these graphs.

Graph 1 shows 300 random numbers that were generated by Microsoft Excel. The numbers range from 0 to 10. For example, the 100^th random number in the series is about 6.3, and the 200^th is about 5.

The Graph 2 is the same as Graph 1 but with the grid lines and scales removed. There are regions with many points, and other regions which are blank. Clusters and blank areas like this are characteristic of random numbers.

Think you can come up with a list of truly random numbers? Think again.

Most people think that consecutive random numbers must be distinctly different, so if asked to make a list of ten random integers ranging from 0 to 9, we might choose 1, 8, 6, 3, 0, 2, 8, 5, 9. But a series like 2, 2, 2, 3, 0, 0, 0, 8, 8, 8 is just as valid and would just as easily show up in a machine-generated list of random numbers. Note that when the first list is fairly evenly distributed through the range of 0 through 9. If our list of 300 numbers was chosen like this, we would see a fairly even distribution of points on Graph 2. But truly random numbers tend to form clusters and blanks like those we see here.

Dynasties and lucky shots: we don't really believe that things happen randomly.

It is likely that when you looked at the Graph 2 you instinctively tried to learn something from the pattern of clusters and blank areas. That is because our survival as human beings depends on the recognition of patterns in nature, and it is why truly random events are difficult for us to accept. We try to predict winning and losing streaks among equally-matched sports teams. We try to understand why disasters seem to occur in twos, or threes. We look for underlying causes of disease outbreaks that seem to appear in one region but not others. We speak of 'the law of averages,' which says that a series of like events must be limited. But random numbers are indeed random, and do not obey any such rules.

How we're fooled by truly random events—we're not as smart as we think:

For example, suppose that each point on Graph 2 represented one case of influenza in an evenly-populated region. You might assume that the people in the blank areas led healthier lifestyles than those in the clusters? But it is also possible that this particular flu strikes people randomly, and we are basing our assumption on the clusters that always occur when we plot random events.

Random number lists prevent collisions of airplanes and e-mail:

We use lists of random numbers for many purposes. Every airliner is equipped with a collision-avoidance system. When two airplanes approach each other, the system on each aircraft communicates with the other and establishes that a potential for collision exists. Then each airplane selects a number from a list of random numbers. The systems then compare numbers. The pilot of airplane whose number is lower is instructed to dive, and that of the airplane with the larger number is told to climb. If the numbers happen to be identical, the two units each select the next number from their lists.

A similar system is used to avoid 'collisions' when more than one computer wishes to use a single shared communications line. If interference is detected, each of the competing machines selects a random number. The machine whose number is lower must wait for the machine whose number is higher.

How to make a list of truly random numbers without guessing:

We use several approaches to generate lists of random numbers. One method uses a mathematical calculation that computes the irrational number p to thousands of decimal places. Then consecutive blocks of these digits are used as the random numbers in a list. For example, 3.14159265358979323846 (p to twenty places) would yield the following list of random numbers: 0.1415, 0.9256, 0.3589, 0.7932, and 0.3846. (Random numbers are usually given as decimals between 0 and 1.) Another method counts the number of electrons that leak through a semiconductor junction during a particular time interval. The random number is that which appears on the device's counter. Many other methods are used by mathematical programs and calculators.

Activity:

Why do you think that people are reluctant to accept that certain events happen randomly instead of in predictable patterns?

Make your own list of random numbers. Sprinkle grains of salt on a piece of graph paper and count the number of grains that fall within each square. Can you think of other methods that might work as well?

Resources:

http://sunny-beach.net/random_numbers/manual/136.htm

http://www.random.org/

http://cis.nci.nih.gov/fact/3_58.htm