# Summary

- Probability is a numerical measure which indicates the chance of occurrence of an event.
- If the probability of occurrence of an event is zero, then it means that the event cannot occur and if the probability is 1, then it means that the event is certain to occur.
**Random experiment:**Process which leads to well-defined results called outcomes; and operation that results into two or more outcomes is called an experiment.**Outcome:**The result of a single trial of a random experiment is called as outcome.**Sample space:**All the possible outcomes of the random experiment form a sample space.**Equally likely events:**Two or more events (outcomes) of an experiment are said to be equally likely if any of them cannot be expected to occur in preference to the other equally likely events are equal probabilities.**Event:**One or more possible outcome of an experiment are said to form an event. Event is the subject of the samples space.**Simple event:**An Event is said to be simple if it corresponds to a single possible outcome of an experiment.**Compound events:**The joint occurrence of two or more simple events is called a compound event.**Mutually exclusive event:**Two events are mutually exclusive if they cannot occur simultaneously in the same trial.P(A âˆ© B) = 0**Exhaustive events**: Events are said to exhaustive if they form a sample space- Sum of all probabilities of events = 1
P(A âˆª B) = 1
**Independent events**: Two events are independent if the occurrence of one does not affect the probability of the other occurring.**Dependent events:**Two events are dependent if the first event affects the outcome or occurrence of the second event in a way the probability is changed**Classical definition**Where,*P*(*A*) < 1- If
*n*(*A*) = 0,*P*(*A*) = 0 - If the event has zero favorable outcomes the event is said to be impossible event
- If
*n*(*A*) =*n*(*s*),*P*(*A*) = 1 *P*(*A*) = 1 means the event is certain

- If
**Complementary event***P*(*A*) = 1 â€“^{c}*P*(*A*)**Statistical/Empirical/Posterior definition of probability***r*= no. of times the event*A*occurs*n*= no. of trials of the experiment- Odds in favour of event
*A*are in ratio*x*:*y* - Odds in against the event
*A*are in ratio*x*:*y* **Addition rule***A*and*B*are any two events then the probability that*at least one*of them occurs is given by*P*(*A*âˆª*B*) =*P*(*A*) +*P*(*B*) â€“*P*(*A*âˆ©*B*)*A*and*B*are mutually Exclusive*P*(*A*âˆª*B*) =*P*(*A*) +*P*(*B*) as*P*(*A*âˆ©*B*) = 0*A*,*B*and*C**P*(*A*âˆª*B*âˆª*C*) â€“*P*(*A*) +*P*(*B*) +*P*(*C*) â€“*P*(*A*âˆ©*B*) â€“*P*(*B*âˆ©*C*)*P*(*C*âˆ©*A*) +*P*(*A*âˆ©*B*âˆ©*C*)*For mutually Exclusive Events**P*(*A*âˆª*B*âˆª*C*) =*P*(*A*) +*P*(*B*) +*P*(*C*)**Conditional probability***P*(*A*) â‰ 0;*P*(*A*) > 0**Multiplication theorem***A*and*B*is given by the product of the unconditional probability of the event*A*by the conditional probability of*B*on the assumption that*A*has occurred.**Independent event***A*and*B*are said to be independent if the occurrence or non-occurrence of one does not affect the occurrence of the other.*P*(*A*âˆ©*B*) =*A*and*B*are independent only if*P*(*A*âˆ©*B*) =*P*(*A*) Ã—*P*(*B*)*f*(*x*) â‰¥ 0 and_{i}**Expected value**- The theoretical mean of the variable
- If Random variable
*x*assumes the value*x*_{1},*x*_{2},*x*_{3 }â€¦*x*with probability_{n}*f*(*x*_{1}),*f*(*x*_{2}),*f*(*x*_{3}) â€¦*f*(*x*). The expected value_{n} *E*(*x*) is also denoted by symbol*Î¼**E*(*x*) =*Î¼*= Î£*px*= mean- Variance
- Standard Deviation
*V*(*x*) =*E*(*x*^{2}) â€“ [*E*(*x*)]^{2}*E*(*k*) =*K*âˆ´*K*is constant*E*(*XY*) =*E*(*X*) Ã—*E*(*Y*) (since*X*and*Y*are independent)