# Definition probability that the coin will land

Definition of Probability :

1.
Probability is a branch of mathematics that deals with counting the
likelihood of a given event’s occurrence, which is expressed as a value between
1 and 0. An event with a probability of 1 can be considered a sureness event  : for example, the probability of a throwing
coin resulting in either “heads” or “tails” is 1, because
there are no other choices, assuming the coin lands flat.

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2.
An
event with a probability of 50% can be considered to have equal odds of
occurring or not occurring: for example, the probability of a coin toss resulting
in “heads” is1/2 , because the toss is equally as likely to result in
“tails.” An event with a probability of 0% can be considered an impossible
: for example, the probability that the coin will land (flat) without either
side facing up is 0%, because either “heads” or “tails”
must be facing up. A few paradoxical, probability theory applies precise
calculations to quantify uncertain measures of random events

Probability
Theorems :

Probability
theory is the branch of mathematics concerned with probability. Although there are several different probability
interpretations, probability theory
treats the concept in a rigorous mathematical manner by expressing it through a
set of axioms. Usually these axioms formalise probability in terms of a probability space , termed
the probability measure, to a set of outcomes called the sample space.

*Terminology for probability
theory:

• experiment: operation of observation or
measurement; e.g., coin flip;

•
outcome:
result acquired through an experiment; e.g., coin
shows tails;

•
sample space: set of all possible results of an experiment; e.g., sample space for
coin flip: S = {H, T}. Sample spaces can be finite or infinite

Often we are interested in integration of two or more events. This
can be represented using set theoretic operations . suppose a sample space S and two events A and B:

•
complement A (also A0 ): all components
of S that are not in A;

• subset
A ? B: all components  of A are also elements of B;

• union
A ? B: all components of S that are in A or B;

•
intersection A ? B: all components of S that are
in A and B.

*Rules of Probability :

1.    Rule of Subtraction: The probability that event A will happen is equal to
1 minus the probability that event A will not happen .

P(A) = 1 – P(A’)

2.    Rule of Multiplication : The probability that Events A and B both happen
is equal to the probability that Event A happens times the probability that
Event B happens, given that A has happened.

P(A ? B) =
P(A) P(B|A)

3.    Rule of Addition: The probability that Event A or Event B happens
is equal to the probability that Event A happens plus the probability that
Event B happens minus the probability that both Events A and B happen .

P(A ? B) = P(A) + P(B)
– P(A ? B)

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