For an event, odds against is $6 : 5$. The probability that event does not occur, is

An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered $2, 3, 4,.......,12$ is picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$, is

For two given events $A$ and $B$, $P\,(A \cap B) = $

In a horse race the odds in favour of three horses are $1:2 ,  1:3$ and $1:4$. The probability that one of the horse will win the race is

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 events such that $P(E)=\frac{1}{4}$,  $P(F)=\frac{1}{2}$ and $P(E$ and $F )=\frac{1}{8},$ find $:$ $P($ not $E$ and not $F)$.