General Probability

1.       The Probability of occurrence of the event E is defined as 

P (E) = n(E)/ n(S)

Where n(E) is no. Of outcomes favourable to occurrence of E and n(S) is all possible outcomes.

e.g.

2.       Odds in favour of an event and odds against an event:

If the no. Of ways in which an event can occur be m and the no. Of ways in which it doesn’t occur be n then,

a.       Odds in favour of an event = m/n

b.      Odds against the event       = n/m

3.  If  ‘E’ is an event, then we consider P(E) as the probability of occurrence of event E.

Let P(E’) is the probability of non-occurence of event E.

P(E’) = 1- P(E)

4. Mutually Exclusive Events

In which occurrence of one event doesn’t depend/effect occurrence of another event. Both (or more) events occur independently. They can’t happen together.

If  E & F are mutually exclusive events, then

a.       P(E and F)  =  0   or     P (E∩F) = 0                       [        ∩   means   “and”        ]

b.      P (E and F)   or   P(EUF)  = P(E) + P(F)

5. Mutually Exclusive Exhaustive events Where occurrence of an event depends/relates to another event.

If  E & F are two mutually exclusive exhaustive events, then

a.       P(E) + P(F) = 1

b.      P(E – F) = P(F) – P(E∩F)

Addition theorem:

P(E or F) = P(E) + P(F) - P(E and F)

Here is the same formula, but using ∪ and ∩:

P(E ∪ F) = P(E) + P(F) - P(E ∩ F)

For any three events E, F and G,

P (E or F or G) = P(E) + P(F) + P(G) – P(E and F) – P(F and G) – P(E and G)

+ P (E and F and G)

6. Conditional Probability

If the events are dependent to each other

Let E and F be the two events associated with the same sample space. Then the probability of occurrence of F under the condition that E has already occurred and P(E) ≠ 0, is called the conditional probability. It is denoted by P(F/E).

Multiplication Theorem

a.       P(E and F) = P(E) × P(F/E)                               where P(E) ≠ 0

b.      Similarly , P(E and F)  = P(F) × P(E/F)            where P(F) ≠ 0

c.       P(F/E)  = n(E and F) ÷ n(E)

d.      P(E/F) = n (E and F) ÷ n(F)

In case if both the events are independent i.e. occurrence of one doesn’t depend on occurrence of another event, then

P(E and F) =P(E) × P(F)