Probability
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| Discuss Probability PROBABILITY
What is Probability?
All of us have sometimes, consciously or unconsciously, compared certain events or future happenings by using the phrases like, ‘likely’, ‘most likely’, ‘probably’, ‘most probably’, etc.. Using these phrases shows that we have some primitive intuition which is helping us to make out the possibility of the occurrence of one event over the other. Probability may be simply defined as the amount of chance favouring the occurrence of that event.
For example, we take the event of the throw of a die. Whenever a die is thrown the probability of any number appearing is 1/6.
Some of the terms related are defined below
MUTUALLY EXCLUSIVE Events are said to be mutually exclusive if no two of then can occur at the same time. Two events A and B are said to be exclusive if they can never occur at the same time i.e A í B is an impossible event => P (A í B) = 0
EQUALLY LIKELY EVENTS Events are said to be equally likely if there is no reason to expect the occurrence of any one in preference to the other
EXHAUSTIVE EVENTS Events are said to be exhaustive if they cover all possible outcomes of the experiment being performed.
FAVOURABLE EVENTS An outcome of an experiment is said to be favourable to n event A if its occurrence implies the occurrence of A. for example, while throwing a dice, if event A means the appearance of an odd number, then face 3 appearing is favourable to A.
DEPENDENT AND INDEPENDENT EVENTS Two events A and B are said to be independent events if the occurrence of one does not depend on the occurrence or non-occurrence of the other.
If m be the number of occurrences of a given event A in a total member of n identical trials, then the ratio m/n is the probability of occurrence of the event A. To be more mathematical, the probability p of an event A is the ratio of the number ‘m’ of favourable cases to the total number of all possible cases in forming a complete group of mutually exclusive, equally likely and exhaustive cases i.e.
P(A) = m/n
NOTE: By definition, any probability has to be between 0 and 1 i.e.
0 < = P (A) < = 1
FUNDAMENTAL THEOREM OF PROBABILITY
For any two events A and B, P (A í B) = P (A)+ P (B) - P (A U B) If A and B are mutually exclusive, then P (A í B) = 0 and so P (A U B) = P (A) + P (B)
CONDITIONAL PROBABILITY
If A and B are dependent events, i.e. the occurrence of A does depend on the occurrence of B, then we define P (A / B), the conditional probability of the occurrence of event A given that B has already occurred.
Example : In the rolling of a die, if event A is appearance of 2 on the top face. Event B is appearance of an even number, then P (A / B) is the probability of appearance of 2 given that an even number has appeared. Since there are only 3 even numbers (2, 4 and 6,) and of them, only appearance of 2 is favourable to A, P (A / B) = 1/3 So P (A / B) = probability of joint occurrence of A and B divided by probability of occurrence of B i.e. P (A / B) = P (A í B) / P (B) similarly P (B / A) = P (A í B) / P (A)
BAYE’S THEOREM
If event A occurs only when one of the events B and C , which form a complete group of exhaustive events, occurs, then the probability of occurrence of A is given by
P (A) = P (B). P (A / B) + P (C) . P (A / C)
Proof :
From above, P (A / B) = P (A í B) / P (B)
Therefore P (A í B) = P (A / B) . P (B)
Similarly, P (A í C) = P (C). P (A / C)
Now if A can occur only if either B or C occur, then
P (A) = P (A í B) + P (A í C) = P (B) P (A / B) + P (C) . P (A / C)
Now, P (B / A) = P (A í B) / P (A) = {P (A / B) . P (B)} / P (A)
Substituting from above
P (B / A) = {P (A / B) . P (B)} / {P (A / B) . P (B) + P (A / C) . P (C)}
similarly, P (C / A) = {P (A / C). P (C)} / {P (A / B). P (B) + P (A / C). P (C)}
This is known as Baye’s theorem.
