Question

If two events are independent, then:
(a) They must be mutually exclusive
(b) The sum of their probabilities must be equal to 1
(c) (a)and(b) are both correct
(d) none of these

Answer 1

For an event to be mutually independent then there should be no relation between both the events, and such events are not interdependent to each other. For example: Throwing a dice and tossing a coin are both events happening somewhere but are not dependent on each other.

Formulae Used: ${P_1}\, + \,{P_2} \ne 1\,or\,\,{P_1}\, + \,P = 1$
For any dependents events ${P_1}\, + \,P = 1$
And for independent events${P_1}\, + \,{P_2} \ne 1\,or\,\,{P_1}\, + \,P = 1$, any case can be followed but it is not must to be
${P_1}\, + \,P = 1$ for any two independent events.

Complete step by step answer:
In the given question we are provided with the information that for any two events to me independent that is there is no connection between the events then what condition they follow:
Let name of two events be $1\,and\,2$
Then probabilities of both events can be written as
${P_1}\,and\,{P_2}$
Now for events to be independent they must be mutually exclusive and
${P_1}\, + \,{P_2} \ne 1\,or\,\,{P_1}\, + \,P = 1$
These both conditions can be satisfied by the events but there is no possibility for:
${P_1}\, + \,P = 1$

Hence, answer of the question is option (a) that is they must be mutually exclusive to be an independent event.

Note: Probability means the chance of occurrence of any desired task in a number of events drawn. The formulae of probability are the ratio of favorable outcome to the total outcome. For example if the probability of coming six in a dice is one by six.