Probability is the measure of the chance of occurrence of a future event. It tells us how likely we expect the event to happen.
Probability of an event occurring \(= \frac{Number of favourable outcomes }{Number of all possible outcomes}\)
Note :
(
a). If an event E is sure to occur, we say that the probability of the event E is equal to 1 and we write \(P (E) = 1\).
(
b). If an event E is sure not to occur, we say that the probability of the event E is equal to 0 and we write \(P (E) = 0\). Therefore for any event E, \(0 ≤ P (E) ≤ 1\)
Mathematical definition of probability :
(A) If the outcome of an operation can occur in n equally like ways, and if m of these ways are favorable to an event E, the probability of E, denoted by \(P (E)\) is given by \(P (E) = \frac{m}{n}\)
(B) As \(0 ≤ m ≤ n\), therefore for any event E, we have \(0 ≤ P (E) ≤ 1\)
(C) The probability of E not occuring, denoted by P(not E), is given by \(P(not E)\) or \( P(E') = 1 – P(E)\)
(D) Odds in favour \(=\frac{ No. of favourable cases }{ No. of unfavorable cases}\)
(E) Odds against \(= \frac{No. of unfavourable cases }{ No. of favorable cases}\)
Mutually Exclusive Events :
Two events are mutually exclusive if one happens, the other can’t happen & vice versa. In other words, the events have no common outcomes. For example
(
a). In rolling a die
E :– The event that the no. is odd
F :– The event that the no. is even
G :– The event that the no. is a multiple of three
(
b). In drawing a card from a deck of 52 cards
E :– The event that it is a spade
F :– The event that it is a club
G :– The event that it is a king
In the above 2 cases events E & F are mutually exclusive but the events E & G are not mutually exclusive or disjoint since they may have common outcomes.Addition law of Probability :
If E & F are two mutually exclusive events, then the probability that either event E or event F will occur in a single trial is given by :
P(E or F) = P(E) + P(F)If the events are not mutually exclusive, then
P(E or F) = P(E) + P(F) – P(E & F together).
Note :
Compare this with n(A ∪ B) = n(A) + n(B) – n(A ∩ B) of set theory. Similarly
P (neither E nor F) = 1 – P(E or F).
Independent Events:
Two events are independent if the happening of one has no effect on the happening of the other.
Example :
1.On rolling a die & tossing a coin together
E :– The event that no. 6 turns up.
F :– The event that head turns up.
2.In shooting a target
E :– Event that the first trial is missed.
F :– Event that the second trial is missed.
In both these cases events E & F are independent. BUT 3. In drawing a card from a well-shuffled pack
E :– Event that first card is drawn
F :– Event that second card is drawn without replacing the first
G :– Event that second card is drawn after replacing the first
In this case E & F are not Independent but E & G are independent.
Multiplication Law of Probability:
If the events E & F are independent then
P(E & F) = P (E) x P (F) & P (not E & F) = 1 – P (E & F together).
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