Two students Anil and Ashima appeared in an examination. The probability that Anil will qualify the examination is 0.05 and that Ashima will qualify the examination is 0.10. The probability that both will qualify the examination is 0.02.
Find the probability that
(a)  Both Anil and Ashima will not qualify the examination.
(b)  Atleast one of them will not qualify the examination and
(c)  Only one of them will qualify the examination.

Answer

(a) 0.87
(b) 0.98
(c) 0.11

Explanation

Let E and F denote the events that Anil and Ashima will qualify the examination,
respectively.

Given that `P(E) = 0.05, P(F) = 0.10` and `P(E nn F) = 0.02`.

(a) The event "both Anil and Ashima will not qualify the examination" may be
expressed as `E' ∩ F'`.

Since, `E'` is ‘not E’, i.e., Anil will not qualify the examination and `F'` is ‘not F’, i.e., Ashima will not qualify the examination.

Also `E' ∩ F' = (E uu F)'` (by De-morgan's Law)

Now `P(E uu F) = P(E) + P(F) – P(E ∩ F)`

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