Probability Problem

Three numbers are chosen uniformly and independently at random from the interval [0,1] and arranged to form an ordered triple (a,b,c) with a<=b<=c. What is the probability that 4a+3b+2c<=1?

I came across this problem while going through some old math materials. Any ideas?

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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`\boxed{123}`	$\boxed{123}$

Comments

I hope this isn't an active Brilliant question. If so, please let me know.

We first consider the volume of the region of $(x,y,z)$ -space for which $0 \le x \le y \le z \le 1$ and $4x + 3y + 2z \le 1$ . This is easy to find: there are four inequalities, which define four planes that enclose a tetrahedron. In the $yz$ -plane (i.e., $x = 0$ ), the given inequalities give the conditions $y \ge 0$ , $z \ge y$ , $3y + 2z \le 1$ . This is a triangle whose vertices are $(x,y,z) \in \{ (0,0,0), (0,\frac{1}{5}, \frac{1}{5}), (0,0,\frac{1}{2})$ . Thus its area is $\frac{1}{20}$ . Now, the height of the tetrahedron as measured from this base is the maximum value of $x$ that satisfies the given constraints--this obviously occurs when $x = y = z$ and $4x + 3y + 2z = 1$ , from which we immediately find $x = \frac{1}{9}$ . Therefore, the total volume of the tetrahedron is $\frac{1}{540}$ .

But this represents the probability that the three numbers chosen satisfy the given conditions without first being reordered in increasing sequence; i.e., that $a = x$ , $b = y$ , and $c = z$ . Hence, the desired probability is $3! = 6$ times the volume found, since there are this many ways to permute three distinct elements--e.g., we could have $a = y, b = z, c = x$ , or $a = z, b = x, c = y$ , etc. Therefore, the desired probability is $\frac{1}{90}$ .

hero p. - 8 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$