Independent Events Complement : 2

An independent event is said to be an event if the outcome of one event does not affect the outcome of the other event. In case of finding the probabilities of the independent events then we need to multiply the probabilities. If two events are said to be mutually exclusive then they are not independent and if two events are independent they are not mutually exclusive.

In this article we are going to see the complement of an independent event.

 

Independent events:

 

Let us consider two events A and B then the probability of both occurring at the same time is given by

                                        P (A and B) = P(A)*P(B)

Ex: Let us consider that we are tossing a coin and rolling a die. Then find the probability of getting the tail side of the coin and the probability of getting three.

Solution:Consider P(A) be the probability of getting the tail while tossing the coin and P(B) be the probability of getting three while rolling a die.

      Then P(A) = `(1)/(2)`  and P(B) =`(1)/(6)`  .

      The probability of the independent events are given by,

      P(A and B) = P(A)*P(B)

                       = `(1)/(2)`  * `(1)/(6)` 

                       = `(1)/(12)` 

 

Complement of independent events:

 

The complement of an independent event (say A) is defined as set of all the outcomes contained in the sample space which are not contained in the outcome of the event. The complement of the independent event is indicated as A

                P(A)=1 – P(`barA` )

 

Ex 1Suppose that we are choosing a card at random from a pack of 52 cards. Find the probability that the card is not a king.

Solution: Let us consider P(A) is the probability of getting the king and P(`barA`)is the probability of not getting the king.

      The complement of an event is given by

                                   P(`barA`) = 1–P(A)

                             Here P(A) = `(4)/(52)` 

                                            = `1-(4)/(52)`           

                                           =  `(48)/(52)` 

                                            =  `(12)/(13) ` 

Ex 2: If we a roll a single dice find the probability of getting a number that is not three.

Solution:Consider P (A) be the probability of getting three and P(`barA`) as not getting three.

                               P(A) = `(1)/(6)` 

                               P(`barA`) = 1– P(A)

                                        = `1-(1)/(6)` 

                                        = `(5)/(6)`