# Probability Exactly One Event Occurs (PDF)

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```					       Probability Exactly One Event Occurs
Probability Exactly One Event Occurs

Probability is mainly define with the help of three basic things or properties that are as
follows:
(a) : There is a set that contains all the possible events that is also known as sample
space Ω.

(b) : There is define one set of possible events that is define as the subset E of
sample space.

(C) : There is also define the probability function P that is applied on the set of all
possible events E.

If we define the probability rules that are expressed using the above symbols as:

(1) : Probability of occurrence any event v is define as P (v) > = 0 Here v is define as
any event that will occur.

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(2) :Total of Probability of all the possible events is P (Ω) = 1.

(3) : There is two events that are p and q then P (p U q) = P (p) + P (q) that define p
and q are disjoint events.

In probability we can define different types of occurrences of events that are define as
the exactly or at most or at least events. So we define the probability exactly one
event occurs as follows:

If there are two events X and Y in the sample space and we want to know the
probability that exactly one of the event X and Y will occur then it will express as:

P (X) + P (Y) – 2 P (X ^ Y).

There is also one other situation is that if X is subset of set Y and the probability of
occurrence of the set Y is P (Y) > 0 then P (X / Y) = P (X) / P (Y).

We can define it by taking an example that is defined as:

Example : If there are given some probabilities that a person randomly buys TV with
probability of 0.6 and computer with probability of 0.3 and probability of buying
nothing is 0.2 then find the probability that the person buy exactly one of the item?

Solution : As we know there are three types of probabilities are given as

Probability of buying TV = 0.6

Probability of buying computer = 0.3

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Probability of buying nothing = 0.2

We can define probability of exactly one item purchase as according to the rule of
exactly one event occur that is P (X) + P (Y) – 2 P (X ^ Y).

So probability of exactly one item purchase = P (TV) + P (Computer) – 2 P ( TV ^
computer)

= 0.6 + 0.3 – 0.2

= 0.9 – 0.2

= 0.7

So the probability that the person buys exactly one of item is 0.7.

We can also take another example to understand the probability of exactly one event
occur as If in a competition there is two questions in which 90 percent of students are
given the answer of first question and 50 percent of students give the both answer
right and 5 percent give the wrong answer of both the questions so find the probability
the exactly one of the answer of questions is right?

Solution : As in the question given

students given the answer of first question = 90 percent = .90

students given the both answer right = 50 percent = .50

students given both wrong answer = 5 percent = .05
So the probability the exactly one of the answer of questions is right = .40 + .05 = .45

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