Definition of Mutually Exclusive Events:
If two events are such that they cannot occur simultaneously for any random experiment are said to be mutually exclusive events.
If X and Y are two mutually exclusive events, then X ∩ Y = ∅
For example, events in rolling of a die are “even face” and “odd face” which are known as mutually exclusive events.
But” oddface” and “multiple of 3” are not mutually exclusive, because when “face3” occurs both the events “odd face” and “multiply of 3” are said to be occurred simultaneously.
We see that two simpleevents are always mutually exclusive while two compound events may or may not mutually exclusive.
Addition Theorem Based on Mutually Exclusive Events:
If X
and Y are two mutually exclusive
events, then the probability of ‘X union Y’ is the sum of the probability of X and
the probability of Y and represented as,
P(X U Y) = P(X) + P(Y)
Proof: Let E be a random experiment and N(X) be the number of frequency of the event X in E. Since X and Y are two mutually exclusive events then;
N(X U Y) = N(X) + N(Y)
or, N(X U Y)/N = N(X)/N + N(Y)/N; Dividing both the sides by N.
Now taking limit N g ∞, we get probability of
P(X U Y) = P(X) + P(Y)
Workedout problems on probability of Mutually Exclusive Events:
1. One card is drawn
from a wellshuffled deck of 52 cards. What is the probability of getting a
king or an ace?
Solution:
Let X be the event of ‘getting a king’ and,
Y be the event of ‘getting an ace’
We know that, in a wellshuffled deck of 52 cards there are 4
kings and 4 aces.
Therefore, probability of getting a king from wellshuffled deck of 52 cards = P(X) = 4/52 = 1/13
Similarly, probability of getting an ace from wellshuffled deck of 52 cards = P(Y) = 4/52 = 1/13
According to the definition of mutually exclusive we know that, drawing of a wellshuffled deck of 52 cards ‘getting a king’ and ‘getting an ace’ are known as mutually exclusive events.
We have to find out P(King or ace).
So according to the addition theorem for mutually exclusive events, we get;
P(X U Y) = P(X) + P(Y)
Therefore, P(X U Y) 
= 1/13 + 1/13 = (1 + 1)/13 = 2/13 
Hence, probability of getting a king or an ace from a wellshuffled deck of 52 cards = 2/13
2. A bag contains 8 black pens and 2 red pens and if a pen is drawn at random. What is the probability that it is black pen or red pen?
Solution:
Let X be the event of ‘getting a black pen’ and,
Y be the event of ‘getting a red pen’.
We know that, there are 8 black pens and 2 red pens.
Therefore, probability of getting a black pen = P(X) = 8/10 = 4/5
Similarly, probability of getting a red pen = P(Y) = 2/10 = 1/5
According to the definition of mutually exclusive we know that, the event of ‘getting a black pen’ and ‘getting a red pen’ from a bag are known as mutually exclusive event.
We have to find out P(getting a black pen or getting a red pen).
So according to the
addition theorem for mutually exclusive events, we get;
P(X U Y) = P(X) + P(Y)
Therefore, P(X U Y) 
= 4/5 + 1/5 = 5/5 = 1 
Hence, probability of getting ‘a black pen’ or ‘a red pen’ = 1
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