How to solve $P(\max\{X_1,X_2,...,X_n\}>b|X_1=a)$ when $X_i$ is uniformly distributed?

Question

Original problem:

Let $X_1,X_2,...,X_n$ be i.i.d. random variables, each uniformly distributed over [0, 1]. Let $V=\max\{X_1,X_2,...,X_n\}$. Determine $P(V>b|X_1=a)$, when $a,b\in [0, 1]$.

My questions:

1. How to evaluate it?

2. $P(V>b|X_1=a)=1-P(X_2\le b)P(X_3\le b)...P(X_n\le b)P(X_1\le b|X_1=a)$, so when $a

Answer 1

If $a>b$ then we already know that the probability of interest is $1$. Otherwise for $a\leq b$,

$$ \begin{align*} P(V>b\mid X_1=a)&=1-P(V\leq b\mid X_1=a) \\ &=1-P(X_1\leq b,\dots,X_n\leq b\mid X_1=a) \\ &=1-\prod_{i=1}^nP(X_i\leq b\mid X_1=a) \\ &=1-\prod_{i=2}^nP(X_i\leq b\mid X_1=a) \\ &=1-\prod_{i=2}^nP(X_i\leq b) \\ &=1-b^{n-1}. \end{align*} $$