Classical Approach:
- If an event can occur in h different ways out of total number of n possible ways, all of which are equally likely, then the probability of the event is (h/n).
Frequency Approach:
- If after n repetitions of n experiments, where n is very large and event is observed to occur in h of these, then the probability of event is (h/n).
- Let S be the Sample Space associated with an experiment E and let A is the event of S and P(A) is the probability of the event A, then the following axioms are satisfied:
- for every event A in S, P(A) >= 0
- for sure and certain events, P(S) = 1
Odds in favor and Odds against favor of an Event:
- If an event can happen in m ways and fails in n ways, then odds in afvor of the event is (m:n) and odds against favor is (n:m), it implies that the probability of success is (m/(m+n)) i.e. P(A) and the probability of failure is (n/(m+n)) i.e. P(A').
- If A and B are two dependent events, then the probability of event A depending on happening of event B is called Conditional Probability.
- If an event can occur in h different ways out of total number of n possible ways, all of which are equally likely, then the probability of the event is (h/n).
Frequency Approach:
- If after n repetitions of n experiments, where n is very large and event is observed to occur in h of these, then the probability of event is (h/n).
Both the classical and frequency approaches were having serious drawbacks. The first because of equally likely and second because of very large numbers.Axiomatic Approach:
- Let S be the Sample Space associated with an experiment E and let A is the event of S and P(A) is the probability of the event A, then the following axioms are satisfied:
- for every event A in S, P(A) >= 0
- for sure and certain events, P(S) = 1
- for the mutually exclusive events A1, A2, ..... An,
P(A1 ∪ A2 ..... An) = P(A1) + P(A2) + ......... P(An).
Mathematical Definition of Probability:
- If in a trial results in n exhaustive, mutually exclusive and equally likely cases and m are the favorable cases, the the probability of happening the event is given by P(E) = (m/n).
Odds in favor and Odds against favor of an Event:
- If an event can happen in m ways and fails in n ways, then odds in afvor of the event is (m:n) and odds against favor is (n:m), it implies that the probability of success is (m/(m+n)) i.e. P(A) and the probability of failure is (n/(m+n)) i.e. P(A').
P(A) + P(A') = 1Conditional Probability:
- If A and B are two dependent events, then the probability of event A depending on happening of event B is called Conditional Probability.
P(A | B) = (P(A ∩ B) / P(B)) P(B | A) = (P(A ∩ B) / P(A))
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