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How can I find the joint probability of the two vectors

How can I find the joint probability of this:

$X_1,X_2 \sim \mathcal{U}(0,1),\quad X_3=1-X_1-X_2\space\space\mathrm{i.e.}\space\space X_3\sim \mathcal{U}(0,1)$

Find the conditional probability:

$$

P(X_1+X_2+X_3\leq t|X_1+X_2+X_3\leq 1)

$$

I tried to use $f_X(x)=f_{X_3}(x|X_1+X_2+X_3\leq 1)f_{X_1,X_2|X_3}(x|X_1+X_2+X_3\leq 1)$ but this did not work so I’m not sure about my attempt.

A:

Hint. Recall that by definition of conditional probability, for fixed $X_1,X_2$, $P(X_3=y|X_1=x_1,X_2=x_2)=f_{X_3|X_1,X_2}(y|x_1,x_2)$. Now, note that by symmetry $f_{X_3|X_1,X_2}(y|x_1,x_2)=f_{X_3|X_1,X_2}(y|x_2,x_1)$.

/*

*

* (C) COPYRIGHT 2014-2015,2017,2018 NAKAMURA KAIZO

*

* SPDX-License-Identifier: GPL-2.0+

*/

#ifndef _ENV_SUPPORT_H_

#define _ENV_SUPPORT_

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