Chapter 1.1 Probability Flashcards
(6 cards)
It is a measure of chance
Probability
What are the definitions/properties of the probability of event A and how is it denoted?
- Non-negativity: States that P(A) must be between the interval [0,1]
- Norming Axiom: States that the probability of the sample space P(Ω) = 1
- Finite Additivity: If A can be expressed as the union of n mutually exclusive events, that is, A = (A1 U A2 U… UAn), then P(A) = P(A1) + P(A2) +…+P(An).
The probability of event A is denoted by P(A).
Fun fact: It was only in the 20th century that Andrey Kolmogrov, a Russian mathematician came up with the acceptable definition of probability which soon became the basis of the modern probability theory
Approaches to assigning probabilities
Assigns probabilities to events before the experiment is performed using the following rule:
If an experiment can result in any one of the N different equally likely outcomes, and if exactly n of these outcomes belong to event A, then P(A) = # elements in A (n)/ # of elements in Ω (N)
Classical probability/approach or a priori
An example of this is the chance of winning the lotto wherein 6 out of 49 numbers will be picked and only 1 of the # of possible combinations
Approaches to assigning probabilities
Assigns probabilities to events by repeating the experiment a large number of times and using the following rule:
If a random experiment is repeated many times under uniform conditions, use the empirical probability of A, then empirical P(A) = # of times event A occured/ # of times experiment was repeated
A posteriori/relative frequency
The a posteriori definition of the the probability of event A is the limiting value of its empirical probability if we repeat the process endlessly.
Example of this is, tossing a coin 4 times and observing the number of heads and tails that come up for 520 times. You will notice that as the more you do an experiment, the more it reaches the approximate of P(A).
It is not restricted to R.Es that generate a sample space that contains equiprobable outcomes (therefore, can be used when the probabilities are not fair/balanced)
Approaches to assigning probabilities
Assigns probabilities to events by using intuition, personal beliefs, and other indirect information
Subjective probability
An example of this is you’re studying for an exam and you don’t feel confident so the chance of you passing (subjectively) is low while another person might think their chances of passing are high. HOWEVER, it is important to note that the measures should still confirm to Kolmogrov’s definitions of a probability and MUST NOT contradict any of the properties of a probability function
