Reflections on school mathematics 
Taking a chance on probability

Jim Stamell, Sylvania High School

Two coloured balls are drawn from a hat ... . Many students and teachers have learnt their probability from textbooks which sport such statements as these. Now while these simplistic contrived situations have their place in the teaching and learning of probability, many texts do not seem to have progressed beyond this. Some students may be forgiven for thinking that probability is all about balls and marbles, and coins and cards, and dice.

In the following examples I have endeavoured to describe some different instances where probability may be used to engender enthusiasm among students. These may be included in existing lessons as a useful extension for more able students. In addition, computer spreadsheets are relied on heavily to produce these results.

The birthday problem

You are in a room with a group of other people. What is the chance that at least two people in the room have the same birthday (day and month only)? How many people would there have to be before we are, say, 50% certain that two people's birthday coincide.

If you are alone in the room, then the probability is certainly zero. If there is another person with you, your chance is 1/366. (For this exercise I'm taking 366 days in the year; would there be much difference if 365 were used?) Perhaps a better approach would be to find the chance of not coinciding and then taking the complement of that number.

With two people in the room the probability is 1 - 365/366. With three people the probability is 1 - 365/366 * 364/366, and so on. With n people (n < 366) the probability that there are at least two persons with coincident birthdays is given by

Obviously, if 366 or more people can be crowded into a room, it is a certainty there will be at least one match.

Table 1 shows the probabilities calculated for up to 50 people. This is then graphed using a spreadsheet program yielding Figure 1. Reading from the graph, it only takes 23 people in the group before there is a greater than 50% chance of this occurring, and 41 people for this chance to become greater than 90%. If the graph is extrapolated to 57 people, the probability reaches 99%.

Figure 1. Chance of two identical birthdays in a group of people

These numbers can be generated on a spreadsheet. Figure 2 shows part of this screen output and the formula typed for cell B4, namely = B3*(367 - A4)/366. The ROUND (  ,3) rounds the answer to three decimal places. The next cell across (C4) is obtained by using =1- B4. Then highlight the required number of cells down, and use FILL DOWN in the Edit Menu to repeat this formula for those cells.

Buffon's needle problem

Georges-Louis LeClerc (1707 - 1788), also known as the Compte de Buffon, is an interesting character. To scientists he was somewhat of an iconoclast, having predicted that the Earth was 75 000 years old as opposed to the then commonly accepted figure of about 6000 years. To mathematicians he stands out first for translating into French, Newton's Method of Fluxions, and second for his needle problem.

If a needle, of length L, is dropped onto a piece of paper which has parallel lines drawn across it spaced A units apart (L<A) (Figure 3) the probability that the needle crosses one of these lines is given by

(If the needle is made half the spacing of the parallel lines, then the probability is reduced to , which can be used to determine the value of .)

Figure 3. Buffon's Needle Problem

A plausible proof of this formula may be given. When a needle is dropped onto a piece of paper it will more than likely fall making an oblique angle (say ) with the parallel lines. Two components of the length of the needle may be taken; one parallel, the other perpendicular to the lines. It is the perpendicular component  that is of interest to us. Reasoning, we can surmise that the probability of this component crossing one of the parallel lines is  Now  may vary between 0 and , so we need to get an 'average' of  for each angle over this domain. We could do this by equating the area under the sine curve with a rectangle  units long and y units wide, and proceed to find the value for y; the 'average' of .

from whence we obtain Putting this in place of  yields the above formula.

Figure 4 is a graph of the probabilities using this formula for differing ratios of L:A. For instance, when a four-centimetre long needle is dropped onto a piece of paper where the line spacings are eight centimetres (L:A = 0.5) the probability that the needle crosses one of the lines is about 0.32 which, in this case, is an approximation for

Figure 4. Buffon's needle problem - Graph of probabilities

Pierre Simon de Laplace (1749-1827) was, with Bernoulli, one of the originators of probability. His work included hydrodynamics, electricity, the study of gravitation, and celestial mechanics. Some texts even argue that the theory of probability owes more to him than to any other single mathematician.

Laplace extended Buffon's needle problem. On the paper he made a criss-cross of two mutually perpendicular sets of equidistant parallel lines. If their distances apart are A and B, and a needle of length 'L' (where L < A, L < B) is allowed to drop onto the paper (Figure 5), the probability that the needle will fall across one of the lines is

(I will leave the proof of this formula to other readers.)

Figure 5. Buffon's needle problem extended by Laplace

It can be shown that when B >> A, this formula reduces to Buffon's formula, as there would now be effectively one set of lines across the paper.

Figure 6 shows graphs of Laplace's extension to Buffon's needle problem for various values of L:A, and for six ratios of A:B. For instance, when a four-centimetre long needle is dropped onto a piece of paper where the line spacings in one direction are eight centimetres
(L:A = 0.5), and twelve centimetres in the other
(A:B = 2.3), the probability that the needle crosses one of the lines is about 0.48. It can also be seen that as B increases relative to A, the graph approaches that in Figure 4.

The idea of central tendency

When we toss a single coin a number of times, we would expect about half of them to come down heads and the other half tails. The operative word here is about. With a large number of flips, say 100, we would be surprised if exactly 50% were heads, even though this would occur more often than any other combination. We would be quite happy with a split such as 45/55 or 58/42 for instance.

If a coin is tossed twice, the probability of half heads is 1/2, but this probability is never again so high, and decreases indefinitely with increasing number of tosses. This is seen in Figure 7 which shows the probability of getting exactly half heads when a coin is tossed an even number of times (up to 100). For six tosses, for instance, the probability is 5/16, and for 1000 tosses it is about . But, by way of contrast, for one thousand tosses of a coin, the chance of scoring somewhere between 480 and 520 heads inclusive is about 80.5%. There are many ways of tossing almost exactly half heads when the number of tosses increases.

Suppose a coin was tossed ten times. We would not be too alarmed if there were 7 heads and 3 tails. The chance of that combination occurring is 11.7%. If the coin was tossed one hundred times, the chance that we get 70 heads and 30 tails is reduced considerably to 0.002 32%. We could intuitively reason that 10 flips is not enough to give us the same sensitivity as 100 flips would.

We could draw graphs showing the probabilities of varying combinations of heads and tails for differing number of tosses (Figures 8, 9, and 10). We could also draw up a table of their main features, including the percentage of tosses that were within, say ±20% and ±10% from equal numbers of heads and tails.

(*Rounded to nearest whole number; there are chances of coin combinations outside 30 heads to 70 heads, but the probabilities of these are )

Figure 8. Single coin flipped 10 times

Figure 9. Single coin flipped 50 times

Figure 10. Single coin flipped 100 times

From this information we can conclude that as the number of coin flips is increased, the chance of getting exactly half heads/half tails gradually reduces, and the chance of getting a coin combination within given limits from this even split increases. In other words, the more often an experiment is repeated, the more likely you are to be within certain limits. This trend is visible from the graphs.

Introducing probability

Laplace wrote that at the bottom line the theory of probability is only common sense expressed in numbers. Now, of course, unlike our students, Laplace was quite adept at mathematics.

When we give our students some practical work in probability in order to get a 'feel' for it, we sometimes devise our own recording sheets, or allow students to keep their own record of wins and losses. I have found the Recording Chart in Figure 11 very useful in this regard. [I am indebted to 'Introducing Probability', from which this chart is copied. This is a DIME Projects resource kit which introduces the topic in an easy hands-on approach without the technical jargon, available from Haese & Harris Publications, 27 Simcock Street, West Beach, SA, 5049.]

Figure 11. Probability Recording Chart

The Recording Chart is devised for 50 trials. From each of the small circles, two lines lead downward; one towards the left (if you lose), the other to the right (for a win). Suppose a student tosses two coins and 'wins' if the coins come down, say, two heads, and otherwise 'loses';. The student would draw in the continuous path from one circle to the other, right or left, accordingly. By zig-zagging down the page in this manner from circle to circle, the student should finally come to the bottom at about 25% wins.

It can be ascertained that the order of wins and losses is immaterial; only the relative frequency of each is relevant. For example, moving from any small circle in the order WWLWLLW will take you to the same small circle as moving in the order LLLWWWW, or any other combination. Besides simplicity, speed of use, and automatic counting, the chart allows students to see trends as the experiments proceed.

And finally ...

The above notes are intended to extend thinking beyond the usual concepts found in school texts and to consideration of interesting issues and trends in probabilty. For students, or even teachers, who wish to reproduce these graphs, a reasonable amount of thinking about probability and the binomial theorem will ensue. Of particular importance will be establishing the mathematical algorithms and translating them for use in computer spreadsheets, and use of these computer spreadsheets as tools in solving mathematical problems, hopefully to come to a better understanding of probability.