Which of the following statements (s) is/ are CORRECT ?

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An event A is known to be independent of events B, B $\cup$ C and B $\cap$ C, then A is also independent of C.

An aircraft is equipped with three engines that operate independently. The probability of an engine failure is 0.01. If only one engine is needed for the successful operation of the aircraft, the probability of successful flight, is 0.999999

After a typist has written ten letters and had addressed then ten corresponding envelopes, a careless maining clerk inserted the letters in the envelops at random, are letter per envelope. The probability that exactly nine letters were inserted in the proper envelops is $\frac{1}{10!}$

Let A^C denotes the complement of an event A. If A and B are two events, then the probability that at most one of A, B occurs is $P (A^{C}) + P (B^{C}) - P (A^{C} \cap B^{C})$

answer is A, B, D.

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Detailed Solution

$\begin{array}{l} 1) Given P (A \cap B) = P (A) p (B) - (1) \\ P (A \cap (B \cup C)) = P (A) \cdot P (B \cup C) - (2) \\ P (A \cap (B \cap C)) = P (A) \cdot P (B \cap C) - (3) \\ To prove that P (A \cap C) = P (A) \cdot P (C) \\ From (2) we get P ((A \cap B) \cup (A \cap C)) \\ = P (A) . {P (B) + P (C) - P (B \cap C)} \end{array}$
$\begin{array}{l} = P (A \cap B) + P (A \cap C) - P (A \cap B \cap C) \\ = P (A) P (B) + P (A) P (C) - P (A) P (B \cap C) \\ = P (A \cap C) = P (A) \cdot P (C) \end{array}$
2)

3) If nine letters are in the correct envelopes, the tenth must be also in the correct, envelope, so the probability is zero
4) Let P(S) = probability of successful flight
Let $P (\bar{S}) =$ probability of unsuccessful flight
$P (S) = 1 - P (\bar{S}) = 1 - 0 .000001 = 0 .999999$