☰ Contents · Algebra

Relative frequency and random variables

Lessons 37–38 · 2 lessons · Sh.A. Alimov, A.R. Xalmukhamedov, M.A. Mirzaakhmedov. Algebra: textbook for Grade 9 of general secondary schools. Revised 4th edition. Tashkent: “O‘qituvchi”, 2019
37

Relative frequency of a random event

Textbook: pp. 194–197
GoalCalculate the relative frequency W(A) = M : N of an event and understand how it is related to probability.
New words
relative frequency · nisbiy chastotafrequency · chastotastatistical probability · statistik ehtimolliklaw of large numbers · katta sonlar qonuni
Explanation

The classical definition needs equally likely outcomes, which are not always known in practice. So experiments are carried out: if event A happens in M of N trials, M is the frequency of A and W(A) = M : N is its relative frequency. As the number of trials grows, the relative frequency settles around some number, which is taken as the statistical probability of the event. For example, in the coin experiments of Buffon and Pearson the relative frequency of heads came out very close to 0.5. Jacob Bernoulli justified this as the law of large numbers: for many trials W(A) ≈ P(A). With few trials, the relative frequency can differ noticeably from the probability.

Worked examples
A footballer scored 28 of 40 penalties in training: M = 28, N = 40, W = 28/40 = 0.7.
225 of 250 seedlings planted in the school garden took root: W = 225/250 = 0.9, i.e. 90%.
Class activity

“Coin series”: each pair tosses a coin 20 times and records the heads; then the whole class’s results are added, W is computed and compared with 0.5.

Practice
1
Event A happened in 20 of 50 trials. W(A) = ?
2
For a large number of trials, what is W(A) close to?
3
In the school canteen 20 of 80 students chose soup. What is the relative frequency of choosing soup?
4
Why can W for heads differ noticeably from 0.5 in only 10 tosses?