Read the following chapters in The Practice of Statistics

• 2003: pgs 488-490
• 2012: NA

The past two units have centered on the concept of probability. What exactly is probability? Consider flipping a fair coin, the probability of a head or a tail is 50%. It is easy to state a probability but in reality how it is interpreted? If a fair coin is tossed 20 times does it mean that the results will yield 10 heads and 10 tails? In actuality there in no guarantee that this will occur. It is possible to flip 20 heads in a row or for that matter 20 tails. Even though the last statement would be a rare occurrence it is possible. So what does the 50% mention above really mean?

To further understand the concept of probability the law of large numbers will be used. A coin can be flipped any number of times with each flip being independent of the other, i.e., the outcome of the next flip is not influenced by the outcome of the previous flip. If N coin flips are performed and N_H represents the number of heads then for any number of flips the ratio of heads to the number of flips is N_H/N.

As N gets larger and larger the expected ratio of N_H/N will get closer to 1/2 or 50%. This is a concept known as a limit in mathematics. As N approaches infinity then the ratio of N_H/N approaches 50%.

Law of Large Numbers

If the probability of a given outcome to an event is P and the event is repeated N times, then the larger N becomes, so the likelihood increases that the closer, in proportion, will be the occurrence of the given outcome to N*P.

In actual practice we cannot flip a coin an infinite number of times so in general this rule applies to situations in which probabilities are assigned to a particular outcome. Also take into account the necessary assumptions when assigning probabilities, e.g., "fair" coin, "fair die, etc.

In other words when we say that the probability of a heads is 1/2, we mean that if we flip the fair coin enough, eventually the number of heads over the number of total flips, N_H/N, will be arbitrarily close to 1/2.

If the probability of throwing a double-6 with two dice is 1/36, then the more times we throw the dice, the closer, in proportion, will be the number of double-6s thrown to of the total number of throws. This is, of course, what in everyday language is known as the law of averages. The overlooking of the vital words 'in proportion' in the above definition leads to much misunderstanding among gamblers. The 'gambler's fallacy' lies in the idea that "In the long run" chances will even out. Thus if a coin has been spun 100 times, and has landed 60 times head uppermost and 40 times tails, many gamblers will state that tails are now due for a run to get even. There are fancy names for this belief. The theory is called the maturity of chances, and the expected run of tails is known as a 'corrective', which will bring the total of tails eventually equal to the total of heads. The belief is that the 'law' of averages really is a law which states that in the longest of long runs the totals of both heads and tails will eventually become equal.

In fact, the opposite is really the case. As the number of tosses gets larger, the probability is that the percentage of heads or tails thrown gets nearer to 50%, but that the difference between the actual number of heads or tails thrown and the number representing 50% gets larger.

Let us return to our example of 60 heads and 40 tails in 100 spins, and imagine that the next 100 spins result in 56 heads and 44 tails. The 'corrective' has set in, as the percentage of heads has now dropped from 60 per cent to 58 per cent. But there are now 32 more heads than tails, where there were only 20 before. The 'law of averages' follower who backed tails is 12 more tosses to the bad. If the third hundred tosses result in 50 heads and 50 tails, the 'corrective' is still proceeding, as there are now 166 heads in 300 tosses, down to 55-33 per cent, but the tails backer is still 32 tosses behind.

Put another way, we would not be too surprised if after 100 tosses there were 60 per cent heads. We would be astonished if after a million tosses there were still 60 per cent heads, as we would expect the deviation from 50 per cent to be much smaller. Similarly, after 100 tosses, we are not too surprised that the difference between heads and tails is 20. After a million tosses we would be very surprised to find that the difference was not very much larger than 20.

A chance event is uninfluenced by the events which have gone before. If a true die has not shown 6 for 30 throws, the probability of a 6 is still 1/6 on the 31st throw. One wonders if this simple idea offends some human instinct, because it is not difficult to find gambling experts who will agree with all the above remarks, and will express them themselves in books and articles, only to advocate elsewhere the principle of 'stepping in when a corrective is due'.

An understanding of the law of the large numbers leads to a realization that what appear to be fantastic improbabilities are not remarkable at all but, merely to be expected.

Probability and chance are all around us. Many TV game shows use chance and probability. The most famous use of probability was used on the game show "Let's Make a Deal". There are many discussions about this show readily available in textbooks and on the internet. So famous it has become to be known as "The Monty Hall Problem", named after the host of the show Monty Hall. As we close this semester you may choose to do a report on the Monty Hall problem for extra credit. The maximum amount of extra credit possible for this assignment is 25 points.

Another interesting "problem" that arises in probability is know as the "Birthday Problem". In this problem the question posed is, "How many people should be gathered in a room together before it is more likely than not that two of them share the same birthday?"

Ignoring the issues of leap years the problem is solved as follows:

When the first person enters the room and announces their birthday, the probability of the second person sharing the same birthday is 1/365. Conversely, the probability of the second birthday being different is the opposite of the first calculation, 364/365. When two birthdays are known, the probability of the third being different is 363/365, as there are now two 'favorable' outcomes among 365. The compound probability of birthday 2 being different from birthday 1, and of birthday 3 being different from the other two, these being independent outcomes, is:

(364/365)*(363/365) = 0.991796 or 99.2% chance that two people will not share the same birthday.

Note the start of the sequence is (365/365). We have removed this as it does not affect the result of the calculation.

All that is necessary now is to continue adding terms to the fraction until it equals less than 1/2 or 50%, since as soon as the probability is less than 1/2 that all birthdays are different, the probability is clearly more than 1/2 that any two are the same. In other words it is more likely than not that two people in the room share the same birthday. The following chart shows the number of the people in the room and the probability that theyDO NOT share the same birthday.

The fraction drops to less than 1/2 with 23 iterations, so it is more likely than not that in any gathering of 23 or more persons, two of them will share a birthday. Only 50 people need be present for the 'coincidence' of two of them having the same birthday to become, roughly, a 30-1 on chance. In a company of 100 employees the odds are more than three million to one on that two share a birthday.

The birthdays proposition is one where a gambler who can estimate probabilities can make money from unsuspecting punters.

You will not be tested over the Monty Hall Problem or the Birthday Problem.

Use the following websites if you need additional help.

• Stattrek - What is Probability?
• Khan Academy - Law of Large Numbers (8:59)