Urn $A$ contains $6$ red and $4$ black balls and urn $B$ contains $4$ red and $6$ black balls. One ball is drawn at random from urn $A$ and placed in urn $B$. Then one ball is drawn at random from urn $B$ and placed in urn $A$. If one ball is now drawn at random from urn $A$, the probability that it is found to be red, is

If $E$ and $F$ are independent events such that $0 < P(E) < 1$ and $0 < P\,(F) < 1,$ then

$P(A \cup B) = P(A \cap B)$ if and only if the relation between $P(A)$ and $P(B)$ is

An electronic assembly consists of two subsystems, say, $A$ and $B$. From previous testing procedures, the following probabilities are assumed to be known :

$\mathrm{P}$ $( A$ fails $)=0.2$

$P(B$ fails alone $)=0.15$

$P(A$ and $ B $ fail $)=0.15$

Evaluate the following probabilities $\mathrm{P}(\mathrm{A}$ fails alone $)$

Two dice are thrown. What is the probability that the sum of the numbers appearing on the two dice is $11$, if $5$ appears on the first