Sample Space

Let $E$ be an event which is neither a certainty nor an impossibility. If probability is such that $P\left(E\right)=1+\lambda +{\lambda }^2$ and $P\left(E^{\text{'}}\right)={\left(1+\lambda \right)}^2$ in terms of an unknown $\lambda$, then $P\left(E\right)$ is equal to

Forty teams play a tournament. Each of them plays with every other team just once. Each game result is a win for one team. If each team has a $50\%$ chance of winnings each game, the probability that at the end of the tournament, every team has won a different number of games is

A bag has $10$ balls. Six balls are drawn in an attempt and replaced. Then another draw of $5$ balls is made from the bag. The probability that exactly two balls are common to both the draw is

If three squares are selected at random from chessboard, then the probability that they form the letter $L$ is

A composite number is selected at random from the first $30$ natural numbers and it is divided by $5.$ The probability that there will be a remainder is

The probability that in a family of five members, exactly two members have birthday on sunday is

South African cricket captian lost the toss of a coin $13$ times out of $14.$ The chance of this happening was 

$A$ draws a card from a pack of $n$ cards marked $1, 2, \dots , n$. The card is replaced in the pack and $B$ draws a card. Then the probability that $A$ draws a higher card than $B$ is

Three balls marked with $1, 2$ and $3$ are placed in an urn. One ball is drawn, its number is noted, then the ball is returned to the urn. This process is repeated and then repeated once more. Each ball is equally likely to be drawn on each occasion. If the sum of the number noted is $6$, then the probability that the ball numbered with $2$ is drawn at all the three occasions, is