Probability MITx 6.041x

Probability MITx 6.041x Notes


Last Updated: September 09, 2021 by Pepe Sandoval



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Introduction to Probability - The Science of Uncertainty

Probability models and axioms

  • Quantitative description: it is a description in terms of numbers

  • A probabilistic model is a quantitative description of a situation, a phenomenon, or an experiment whose outcome is uncertain.

  • Creating a model involves the following steps:
    1. Define the sample space: Describe the possible outcomes of the experiment (E.g. flip of a coin, roll a dice, etc.), composed of
      • Sample spaces are sets which can be discrete, finite, infinite, continuous (recorded with infinite precision)...
      • List (set) of the possible outcomes ($\Omega$), the elements of the set must be:
        • Mutually Exclusive: At end of experiment there can only be one of the outcomes of the set
        • Collectively Exhaustive: Together all the elements of the set exhaust all the possibilities, at the end of the experiment you should be able to select one from the set
        • At the 'right' Granularity includes only the different outcomes in what we are considering relevant and not different in irrelevant outcomes. E.g. in the flip a coin example the sample space (set) would be Heads=0 or Tails=1, instead of Heads & Rain=0, Heads & No Rain=1, Tails & Rain=2, Tails & No Rain=3, whether is raining it's irrelevant, we only include what is relevant
    2. Specify a probability law which assigns probabilities to outcomes or to collection of outcomes, the probability law specifies which outcomes are more likely to occur and which are less likely
      • Probability is usually assigned to events (an event is a subset of the sample space). We assign probabilities to the various subsets of the sample space.
      • Probabilities are given between 0 (practically cannot happen) and 1 (practically certain event will happen)
      • Probabilities have certain axioms or basic properties (axioms: unprovable rule or principle accepted as true) for example probabilities cannot be negative.
      • Axioms of probability:
        • Non-negativity: Probabilities are positive $P(A) \ge 0$
        • Normalization: We are certain event Omega will occur $P(\Omega) = 1$
        • Additivity The probability that the outcome of the experiments falls in Event A or B is equal to the sum of the probabilities of these two sets.

Axioms of probability

  • There are probabilistic experiments that can be described in stages (E.g two rolls of a tetrahedral die), these type of experiments can be represented using a sequential tree or description

probabilistic sequential tree

Properties of Probabilities

properties of probabilities

  • if $A$ is subset of $B$ the probability that event $B$ occurs ($P(B)$) must be greater or equal than the probability that event $A$ occurs

$$A \subset B \Rightarrow P(A) \le P(B)$$

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

properties of probabilities pt. 2

  • Discrete Uniform Probability Law
    • A finite sample space consist of $n$ equally likely elements
    • The probability that event $A$ occurs which can be set of $k$ outcomes is $P(A) = \dfrac{k}{n}$

properties of probabilities pt. 2

  • Probability calculations steps:
    • Specify sample space (values of the outcomes)
    • Specify probability law
    • Identify the probability of an event of interest
    • Calculate

References

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