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Introductory Probability Concepts Explained
Here are some brief definitions of some introductory probability concepts:
Discrete Probability Distributions: Is a type of probability distribution where a given random variable can only take on values defined in a specified set.
- An example of a discrete probability distribution is the classic coin toss experiment where the only two possible outcomes are either heads or tails.
Continuous Probability Distributions: Is a type of probability distribution where a given random variable x can take on a potentially infinite number of possible values. Accordingly, the probability associated with any specific value of a continuous distribution is null. This means that continuous distributions can only be described in terms of probability density. This density can be converted into the probability that a value will fall within a certain range.
- An example of a continuous probability distribution is the calculation that a given man is within 65 to 70 inches in terms of height. The output of this calculation would be a numerical and discrete probability, but the man's exact height will never be known.
Discrete Uniform Distribution: In a set of integer values that occur in the same frequency, all the integers 1 through k(where k is the last integer in the set) must occur with equal probability. This is known as Discrete Uniform Distribution.
Conditional Probability: Is the probability of event A occurring given that event B has ocurred. Or put more generally, it is the likelihood of an outcome occurring based on the occurrence of a previous outcome.
- The example defined above can be written mathematical as
P(A | B). - From here, if the events A and B are independent then the conditional probability of A given B works out to just the probability of B(
P(B)). - If the events A and B are not independent then the conditional probability of A given B can be derived by solving the following equation:
P(A | B) = P(B ∩ A) / P(B)
- The example defined above can be written mathematical as
Probability Axioms: These are some presupposed true statements(which is the meaning of the term 'axiom') about probability. Let's quickly go over three probability axioms:
- Probability of Event: This axiom states that the probability of any event is always between 0 and 1. The probability can also be either 0 or 1. This probability is derived by dividing the number of outcomes of the specific event by the total number of outcomes in the sample space
- Probability of Sample Space: This axiom states that given multiple different events that have unique probabilities, when we add up all the probability values of all the events, the final result should equal 1. This value of 1 is the probability for the entire sample space.
- Mutually Exclusive Events: This axiom states that two given events, let's say A and B, cannot occur together. Differently stated, their intersection is null. Note that although A and B are mutually exclusive from each other, they are not complements of each other
Multiplication Rule of Probability: Is a rule that states: The probability of some events C and D occurring is equal to the product of the probability of D occurring and the conditional probability of C given D occurring.
- Here is the mathematical formula for it:
P(C ∩ D) = P(D) . P(C|D)
- Here is the mathematical formula for it:
Addition Rule of Probability: Is a rule that states: The probability of two events, say C and D, is the sum of those two events minus the probability of both events happening.
- Here is the mathematical formula for it:
P(C ∪ D) = P (C) + P(D) - P(C ∩ D)
- Here is the mathematical formula for it:
Conclusion
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