Suppose that all the four possible outcomes `e_1`, `e_2`, `e_3`, `e_4` of an experiment are equally likely. Define the events A, B, C as `A = (e_1, e_4)`, `B = (e_2, e_4)`. What can you say about the dependence or independence of events A, B, C?

Suppose that all the four possible outcomes e_(1),e_(2),e_(3) and e_(4) of an experiment are equally likely. Define the events A, B, C as A={e_(1), e_(4)}, B={e_(2),e_(4)}, C={e_(3), e_(4)} What can you say about the dependence or independence of the events A, B and C ?

Prove that if E and F are independent events, then so are the events E and F'.

If E and F are the complementary events of events E and F , respectively, and if P(F) in [0,1]

Suppose y=e^(a x)+e^(b x) , find y''

A die is thrown. If E is the event the number appearing is a multiple of 3' and F be the event 'the number appearing is even' then find whether E and F are independent ?

Suppose E_(1), E_(2), E_(3) be three mutually exclusive events such that P(E_(i))=p_(i)" for " i=1, 2, 3 If p_(1), p_(2), p_(3) are the roots of 27x^(3) -27x^(2)+ax-1=0 the value of a is -

A ship is fitted with three engines E_1,E_2 and E_3 . The engines function independently of each other with respective probabilities 1/2, 1/4, and 1/4 . For the ship to be operational at least two of its engines must function. Let X denote the event that the ship is operational and let X_1, X_2 and X_3 denote, respectively, the events that the engines E_1, E_2 and E_3 are function. Which of the following is/are true? (a) P(X_1^c "|"X)=3/(16) (b) P (exactly two engines of the ship are functioning "|"X ) =7/8 (c) P(X"|"X_2)=5/6 (d) P(X"|"X_1)=7/(16)

A ship is fitted with three engines E_1,E_2 and E_3 . The engines function independently of each other with respective probabilities 1/2, 1/4, and 1/4 . For the ship to be operational at least two of its engines must function. Let X denote the event that the ship is operational and let X_1, X_2 and X_3 denote, respectively, the events that the engines E_1, E_2 and E_3 are function. Which of the following is/are true? (a) P(X_1^c "|"X)=3/(16) (b) P (exactly two engines of the ship are functioning "|"X ) =7/8 (c) P(X"|"X_2)=5/6 (d) P(X"|"X_1)=7/(16)

A ship is fitted with three engines E_1,E_2 and E_3 . The engines function independently of each other with respective probabilities 1/2, 1/4, and 1/4 . For the ship to be operational at least two of its engines must function. Let X denote the event that the ship is operational and let X_1, X_2 and X_3 denote, respectively, the events that the engines E_1, E_2 and E_3 are function. Which of the following is/are true? (a) P(X_1^c "|"X)=3/(16) (b) P (exactly two engines of the ship are functioning "|"X ) =7/8 (c) P(X"|"X_2)=5/6 (d) P(X"|"X_1)=7/(16)

An unbiased normal coin is tossed n times. Let E_1: event that both heads and tails are present in n tosses. E_2: event that the coin shows up heads at most once. The value of n for which E_1a n dE_2 are independent is _________.