Mutually Exclusive Events

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” odd-face” and
“multiple of 3” are not mutually exclusive, because when “face-3” occurs both
the events “odd face” and “multiply of 3” are said to be occurred
simultaneously.

We see that two
simple-events 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)

Worked-out problems on probability of Mutually Exclusive Events:

1. One card is drawn
from a well-shuffled 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 well-shuffled deck of 52 cards there are 4
kings and 4 aces.

Therefore, probability
of getting a king from well-shuffled
deck of 52 cards = P(X) = 4/52 = 1/13

Similarly, probability of getting
an ace from well-shuffled deck of 52
cards = P(Y) = 4/52 = 1/13

According to the
definition of mutually exclusive we know that, drawing of a well-shuffled 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)

Hence, probability of getting a king or an ace from a well-shuffled 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)

Hence, probability of getting ‘a black pen’ or ‘a red pen’ = 1