Probability MITx 6.041x Notes

Last Updated: September 09, 2021 by Pepe Sandoval

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*Quantitative description*: it is a description in terms of numbersA

**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:
**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

**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.

- Probability is usually assigned to

- 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

- 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

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

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

**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 $P(A) = \dfrac{k}{n}$

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

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