If $E_1, E_2, \ldots, E_n$ are independent events such that $P(E_r) = \frac{1}{1+r}$ for $r = 1, 2, \ldots, n$,then the probability that at least one of $E_1, E_2, \ldots, E_n$ happens is

Two players $A$ and $B$ are alternately throwing a coin and a die together. $A$ player who first throws a head and a $6$ wins the game. If $A$ starts the game,then the probability that $B$ wins the game is:

If two numbers $p$ and $q$ are chosen randomly from the set $\{1, 2, 3, 4\}$,one by one,with replacement,then the probability of getting $p^2 > 4q$ is

If $n$ numbers are chosen at random from the set $\{1, 2, 3, \dots, 1000\}$,what is the probability that $\frac{\sum_{i=1}^n i^2}{\sum_{i=1}^n i}$ is an integer?

$A$ and $B$ are independent events with $P(A)=\frac{1}{4}$ and $P(A \cup B)=2 P(B)-P(A)$,then $P(B)$ is

Choose a number $n$ uniformly at random from the set $\{1, 2, \ldots, 100\}$. Choose one of the first seven days of the year $2014$ at random and consider $n$ consecutive days starting from the chosen day. What is the probability that among the chosen $n$ days,the number of Sundays is different from the number of Mondays?