Frequency and Probability of a Random Event
We often need to conduct observations, experiments, or trials. Such studies frequently result in outcomes that cannot be predicted in advance.
For example, if you open a book at random, you cannot know in advance which page number you will see. It is impossible to predict the final score of a football match before it begins.
As a rule, an observation or experiment is determined by a certain set of conditions. The result of an observation, trial, or experiment is called an event.
A random event is a result of an observation or experiment that, under a given set of conditions, may or may not occur.
Frequency of a Random Event
The frequency of a random event is the ratio:
Statement. Demographers are familiar with the number 0.512. Statistical data from different times and countries show that for every 1000 newborns, on average 512 are boys.
Solution. The number 0.512 is the frequency of the random event “birth of a boy.” In such cases we say that the probability of “birth of a boy” is approximately 0.512.
Statistical Stability of Frequency
To become more familiar with the concept of probability of a random event, let us consider the classic example of coin tossing.
Since the 18th century, many researchers have conducted series of coin-tossing experiments. The table below shows results from several such experiments.
| Researcher | Number of tosses | Number of heads | Frequency |
|---|---|---|---|
| Georges-Louis Leclerc de Buffon (1707—1788) | 4,040 | 2,048 | 0.5069 |
| Augustus De Morgan (1806—1871) | 4,092 | 2,048 | 0.5005 |
| William Stanley Jevons (1835—1882) | 20,480 | 10,379 | 0.5068 |
| Vsevolod Romanovsky (1879—1954) | 80,640 | 39,699 | 0.4923 |
| Karl Pearson (1857—1936) | 24,000 | 12,012 | 0.5005 |
| William Feller (1906—1970) | 10,000 | 4,979 | 0.4979 |
The data reveals a clear pattern: with many coin tosses, the frequency of heads deviates only slightly from 0.5.
The more trials we conduct, the more accurate the estimate of the probability of a random event based on its frequency becomes.
Such an estimate of probability is called statistical. It is used in many fields of human activity: physics, chemistry, biology, insurance, sociology, economics, healthcare, sports, and more.
Probability of an Event
The probability of an event is denoted by the letter (the first letter of the French word probabilite — probability). If in the first example we denote the event “birth of a boy” by the letter , then the result is written as:
Given the approximate nature of the statistical estimate, we can write or .
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From a large batch of light bulbs, 1000 were selected, among which 5 turned out to be defective. Estimate the probability of buying a defective bulb.
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During a flu epidemic, 7,900 out of 40,000 examined residents were found to be ill. Estimate the probability of the event “a randomly selected person has the flu.”
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A help desk operator talks on the phone for an average of 6 hours during a workday (9:00—17:00). Estimate the probability that the phone will be free when you call the help desk during this period.