The unit interval $(0,1)$ is divided at a point chosen uniformly distributed over $(0,1)$ in $\mathbb{R}$ into two disjoint subintervals. The expected length of the subinterval that contains $0.4$ is ________. (rounded off to two decimal places)

Let $X \sim \text{Uniform}(0, 1)$ be the random variable representing the division point in the interval $(0, 1)$.

The two disjoint subintervals created are $(0, X)$ and $(X, 1)$.

The point $0.4$ belongs to:

• $(X, 1)$ with length $1 - X$, if $X \le 0.4$.

• $(0, X)$ with length $X$, if $X > 0.4$.

Let $L$ be the length of the subinterval containing $0.4$:

Since $X$ has a probability density function $f_X(x) = 1$ for $x \in (0, 1)$, the expected value $E[L]$ is:

Evaluating the integrals:

Summing the two parts:

Thus, the expected length rounded off to two decimal places is 0.74 (acceptable range: 0.70 to 0.80).