Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all previous coin tosses. Let $E_1$ be the event that among the second, fourth, and sixth coin tosses, there are at least two heads. Let $E_2$ be the event that among the first, second, third, and fifth coin tosses, there are equal number of heads and tails.

The conditional probability $P\left(E_1 \mid E_2\right)$ is equal to $\_\_\_\_$ . (rounded off to one decimal place)

Let $X$ be a random variable which takes values in the set $\{1,2,3,4,5,6,7,8\}$.

Further, $\operatorname{Pr}(X=1)=\operatorname{Pr}(X=2)=\operatorname{Pr}(X=5)=\operatorname{Pr}(X=7)=\frac{1}{6}$ and $\operatorname{Pr}(X=3)=\operatorname{Pr}(X=4) =\operatorname{Pr}(X=6)=\operatorname{Pr}(X=8)=\frac{1}{12}$.

The expected value of $X$, denoted by $E[X]$, is equal to $\_\_\_\_$ . (rounded off to two decimal places)

A quadratic polynomial $(x-\alpha)(x-\beta)$ over complex numbers is said to be square invariant if $(x-\alpha)(x-\beta)=\left(x-\alpha^2\right)\left(x-\beta^2\right)$. Suppose from the set of all square invariant quadratic polynomials we choose one at random.

The probability that the roots of the chosen polynomial are equal is (rounded off to one decimal place)

The unit interval $(0,1)$ is divided at a point chosen uniformly distributed over $(0,1)$ in $R$ into two disjoint subintervals.

The expected length of the subinterval that contains 0.4 is _________ . (rounded off to two decimal places)