Abstract
The concept of probability distribution is one of the most relevant but, in some ways, among the most complex concepts to be treated at school. If we exclude some fairly simple cases, there is a variety of distributions whose formulation and use is not trivial. In this chapter, many distributions have been omitted (e.g., exponential, t student, Weibull, etc.) because they are too specialized and reserved for a higher level of study. The choice has privileged distributions that can be analyzed starting from objects with which students are, or maybe, familiar. However, the distributions addressed are sufficient to give a fairly broad idea of the topic.
TopIntroduction
Isn't it extraordinary how difficult it is to get a sample really random? ~ George Yule
A premise: “mathematical knowledge in the classroom is no professional and disciplinary knowledge. […]. School mathematics is not abstract and universal, but local and exemplary”. (Steinbring, 1984).
This is one of the reasons that guided the choice of topics.
A probability distribution, a term already encountered in the text, is a mathematical model that links the values of a variable to the probabilities that these values can be observed. The first distribution that is normally dealt with in didactics is the Binomial Distribution. The coin toss is often brought as an example of the probability of obtaining k success in n independent tests according to Bernoulli's well-known formula:
An example of a different and quite particular application is the following: the same type of engine equips two aircraft: a twin-engine and a four-engine. The twin-engine flies even with only one engine, the four-engine flies even with only two engines. Calling p the probability of an engine failure, which is the safest aircraft? Let's assume that the failure of one engine does not affect the operation of the others and that it is therefore an independent event. In this case we have two probability functions that depend on an independent variable that is in turn a probability (of engine failure).
The probability of remaining in flight for the two types of aircraft can be determined by considering the possible combinations. The twin-engine aircraft will arrive at its destination if at least one of its engines is functioning:
P(twin-engine) = C
2,1 p (1-p) + C
2,2 (1-p)
2 = 1 - p
2The aircraft equipped with four engines arrives at its destination if the failures do not involve more than half of the engines:
P(four-engine) = C
4,2 p
2 (1-p)
2 + C
4,3 p (1-p)
3 + C
4,4 (1-p)
4 = 1 - 4p
3 + 3p
4The graph (Figure 1) corresponding to the different values that the breaking-point probability could assume shows clearly that there is a breaking point where the probabilities are equal. All this is obviously unrealistic because the probability of failure is extremely low, especially when compared with the number of car and rail accidents due to breakdowns.
Figure 1. Comparison between twin engine and four engine
Based on the scale of the variable of interest, two types of probability distributions can be distinguished:
- 1.
discrete distributions: the variable is measured with integer numerical values, e.g. the number of people served by a supermarket checkout over a certain period or the longevity of a person expressed in years;
- 2.
continuous distributions: the variable can assume a theoretically infinite number of values within a finite or infinite range, for example the heights of a group of people or the duration of operation of an electronic component.
TopLottery
A lottery is a form of gambling based on the extraction of numbers, with cash prizes. It is one of the games with an ancient historical tradition. The common aspect of the various lotteries is that in the past they have often been used to finance public works. Traces of this game date back to the Han Dynasty in China, around 200 BC. Already in use in Imperial Rome, it partially declined in the Middle Ages. In 1500 the king of France, Francis I, officially legalized the lotteries with an edict, shortly after these were authorized by Queen Elizabeth I. A lot of news can be obtained from a text, not recent, by Alan Wykes. Currently the countries that manage this activity are distinguished by the different ways of execution and therefore it is not easy a unitary treatment. The common aspect is the extraction of numbers using some form of device (Figure 2).
Key Terms in this Chapter
Standard Normal Distribution: Is obtained by a transformation of variable X of a normal curve.
Binomial Distribution: Discrete distribution of the number of successes in a sequence of independent experiments yes-no.
Geometric Distribution: Distribution of the number of failures before the first success.
Poisson Distribution: Discrete distribution expressing the probabilities for the number of events occurring successively and independently in a given time interval.
Normal Distribution: The most famous and used of the continuous distributions.
Rectangular Distribution: Continuous distribution attributing the same probability to all points belonging to a given range.
Large Number of Rare Events: Frequency distribution estimation when rare events are common.
Uniform Distribution: Discrete distribution attributing the same probability to all points belonging to a given range.
Singular Distribution: Assumes values only in particular measurement sets.
Hypergeometric Distribution: Discrete distribution describing the extraction without reinsertion of certain objects.