Question

Consider the following statements
Statement I Two independent events having non-zero probabilities of occurrence cannot be mutually exclusive, vice-versa is also true.
Statement II Two experiments having events $E$ and $F$ respectively are said to be independent, if $P(E \cap F)=P(E) \cdot P(F)$.
Choose the correct option.

• A. Statement I is true
• B. Statement II is true
• C. Both statements are true
• D. Both statements are false

Answer 1

Answer: (C)

I. Term 'independent' is defined in terms of 'probability' of events whereas mutually exclusive is defined in term of events (subset of sample space). Moreover, mutually exclusive events never have an outcome common, but independent events, may have common outcome. Clearly, 'independent' and 'mutually exclusive' do not have the same meaning. In other words, two independent events having non-zero probabilities of occurrence cannot be mutually exclusive and conversely, i.e., two mutually exclusive events having non-zero probabilities of occurrence cannot be independent.
II. Two experiments are said to be independent if for every pair of events $E$ and $F$, where $E$ is associated with the first experiment and $F$ with the second experiment, the probability of the simultaneous occurrence of the events $E$ and $F$ when the two experiments are performed is the product of $P(E)$ and $P(F)$ calculated separately on the basis of two experiments, i.e., $P(E \cap F)=P(E) \cdot P(F)$.