Mutually Exclusive Events
Events of random experiment are said to be, mutually exclusive
, if the happening of one event, prevents the happening of other events.
For example, when two teams A and B are playing a game, the events, "A winning the game", and "B winning the game are mutually exclusive. When A,B,C,D are appearing for an examination, the event "A passing in the examination" does not prevent the events of B,C or D,passing in the examination.
Hence these events are not mutually exclusive.
Definition:
In probability , events E1, E2, ..., En are said to be mutually exclusive if occurrence of any one of them automatically causes the non-occurrence of the remaining n 1 events.
(or)
Any two events of a random experiment are called mutually exclusive if E1 E2 = 'phi'
Example: Consider a random experiment of tossing a coin and rolling a die simultaneously.
The sample space = {(H,1),(H,2), (H,3),(H,4),(H,5),(H,6),(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)}
If E1 = {(H,1),(H,2), (H,3)} ,
E2 = {(H,4),(H,5),(H,6)} and
E3 = {(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)}
E1 E2 ='phi'
E2 E3 = 'phi'
' Therefore ,' E1, E2 are Mutually exclusive
E2, E3 are also mutually exclusive
The probability of happening of two or more mutually exclusive events is equal to the summation of the probabilities of the individual events.
Theorem: If E1 and E2 are mutually exclusive then P(E1'uu' E2) = P(E1) + (E2)
Proof: From addition theorem of probability we have,
P(E1'uu' E2) = P(E1) + (E2) P(E1'nn' E2)
And from the definition of mutually exclusive,
P(E1'nn' E2) = 'phi'
Therefore P(E1'uu' E2) = P(E1) + (E2)
Hence the proof.
Ex : When we throw a die once the occurrence of 5 or 6, but not both, in the same toss.
p = p1 + p2
= '(1)/(6)' + '(1)/(6)'
='(1)/(3)'
Some more Examples of Mutually Exclusive Events
If A,B,C are three horses in a race ,then the events of A,B,C to win the race are mutually exclusive events.
The two events it rains on today and it doesnt rain today are the two mutual exclusive events.
Students can also utilize the
sample question papers for class 9 cbse available online for the reference.
Solved Example:
The probabilities of three teams P, Q and R winning a game competition are '(1)/(2)' , '(1)/(3)' , '(1)/(8)' Calculate the probability that
a) either P or Q will win
b) either P or Q or R will win
c) none of these teams will win
d) neither P nor Q will win
Sol: a) probability that either P or Q will win ='(1)/(2)' + '(1)/(3)' = '(5)/(6)'
b) probability that either P or Q or R will win ='(1)/(2)' + '(1)/(3)' + '(1)/(8)' = '(23)/(24)'
c) probability that none of these teams will win = 1 P(A or B or C will win) = 1- '(23)/(24)' = '(1)/(24)'
d) probability that neither P nor Q will win = 1 P(either A or B will win) = 1- '(5)/(6)' = '(1)/(6)'
Exercise Problems on Mutually Exclusive Events
1. Find the probability of occurring a head or tail when a coin is tossed.
2.Find the probability that a die when rolled it will show 3 or 5.
by: nitinp
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