PROBABILITIES Course

Calculate a discrete probability

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In discrete probabilities, both A and Omega are finite sets, so in most cases, all you have to do is to evaluate

@ \mathbb{P}(A) = \color{green}{\frac{card(A)}{card(\Omega)} = \frac{|A|}{|\Omega|}} @

You must evaluate

  • $card(A)$: how many values are in A, your event
  • $card(\Omega)$: how many values are in Omega

Then the probability is a sort of "percent" of success by the number of tries. But that's a value in $[0,1]$ since we are not multiplying it by 100.


Uniform probability

Sometimes, the probability is called uniform probability because each event has the same probability of happening.

That's the case for a normal dice, you got the same probability of having one of the 6 values: $\frac{1}{6}$. That's also the case in a card game, like $\frac{1}{52}$ if you got 52 cards.

The formula is

@ \forall{w} \in \Omega \ then \ \mathbb{P}(w) = \frac{1}{card(\Omega)} @


Conditional probability

You may have to calculate a probability like these

  • Knowing that B happened, what's the probability of A happening?
  • Given B, what's the probability of A??
  • Sachant B, quelle est la probabilité de A?

If that's the case, then that's a conditional probability.

@ \mathbb{P}(A|B) := \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} @

$\mathbb{P}(A|B)$ may also be written $P_B(A)$ or $P(B \wedge A)$ (rarely used).

Properties

  • $\mathbb{P}(\overline{A}|B) = 1 - \mathbb{P}(A|B)$
  • $\mathbb{P} (B \cup C|A) = \mathbb{P}(B|A) + \mathbb{P}(C|A) − \mathbb{P}(B ∩ C|A)$

Bayes formula

Also called probabilité des causes, using the previous formula, you could deduce

@ \mathbb{P}(A \cap B) = \mathbb{P}(B) * \mathbb{P}(A|B) = \mathbb{P}(A) * \mathbb{P}(B|A) @

And we could write

@ \mathbb{P}(A|B) := \frac{\mathbb{P}(B) * \mathbb{P}(A|B)}{\mathbb{P}(B)} \ or \ \frac{\mathbb{P}(A) * \mathbb{P}(B|A)}{\mathbb{P}(B)} @

This kind of probability is called Posterior probability (a posteriori, wiki) as the opposite of A priori probability (a priori, wiki).


Law of total probability

In French, it's called Formule des probabilités totales.

\[ \begin{split} \mathbb{P}(B) = \mathbb{P}(B \cap \Omega) = \mathbb{P}((A_1 \cap B) \cup (A_2 \cap B) \cup \ldots) \\ = \mathbb{P}(A_1 \cap B) + \mathbb{P}(A_2 \cap B) + \ldots \\ = \mathbb{P}(A_1) * \mathbb{P}(B|A_1) + \mathbb{P}(A_2) * \mathbb{P}(B|A_2) + \ldots \end{split} \]

Chain rule

In French, it's called Formule des probabilités composées. It's defined by

\[ \prod_{i=1}^n \mathbb{P}(A_i | \bigcap_{j=1}^{i-1} A_j) \]

For instance if $n=4$, we got

\[ \mathbb{P}(A_4 \cap A_3 \cap A_2 \cap A_1) = \mathbb{P}(A_1) * \mathbb{P}(A_2|\ A_1) * \mathbb{P}(A_3|\ A_2 \cap A_1) * \mathbb{P}(A_4|\ A_3 \cap A_2 \cap A_1) \]