written by Rustam Fayzullayev

📌 Assessment

What is Probability?

flowchart LR
    I["0<br>Impossible"] --- H["0.5<br>Even chance"] --- C["1<br>Certain"]
    style I fill:#f4b6b6
    style H fill:#f7e3a1
    style C fill:#9ed49a

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Probability is the numerical measure of the likelihood that an event will occur. It is always a number between 0 and 1 (or 0% and 100%).

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Rus: Вероятность – числовая мера возможности наступления события.

$$ 0 \le P(A) \le 1 \quad \text{for any event } A $$

  1. Certain event – sure to occur, probability = 1
  2. Impossible event – has no chance of occurring, probability = 0
  3. Mutually exclusive and collectively exhaustive events – their probabilities add up to 1: P(A) + P(B) + P(C) = 1

📌 A probability can never be negative and can never be above 1. If a question gives you P = 1.2 or P = −0.1, something is wrong.

Approaches to Probability

flowchart TB
    P["Probability"] --> C["CLASSICAL<br>equally likely outcomes"]
    P --> E["EMPIRICAL<br>relative frequency of past trials"]
    P --> S["SUBJECTIVE<br>personal judgement"]
  1. Classical – based on the assumption that all outcomes are equally likely, e.g. rolling a 4 with a fair 6-sided die: P = 1/6
  2. Empirical – the number of times an event occurred as a proportion of a known number of trials, e.g. 650 of 10,000 loans defaulted in the last 5 years: P(default) = 650 ÷ 10,000 = 0.065
  3. Subjective – assigned by an individual based on whatever information is available, e.g. the probability that the Uzbek national football team reaches the knockout stage of the 2030 World Cup

Rus: Классическая, эмпирическая (статистическая, частотная) и субъективная вероятность.

$$ \text{Classical: } P(A) = \frac{\text{number of outcomes in } A}{\text{total number of outcomes}} \qquad \text{Empirical: } P(A) = \frac{\text{times } A \text{ occurred}}{\text{number of trials}} $$