Probability about three independent exponential random variables Suppose we have three independent exponential random variables $A$, $B$ and $C$  with respective parameters $1$, $2$ and $3$. 

Calculate $P(A&lt;B&lt;C)$.

The hint says this problem could be solved with calculus and without calculus. I am really curious how to approach it with different methods.
It is rather well known that $$P(A&lt;B)=\frac{\lambda_a}{\lambda_a+\lambda_b}.$$

You should also know the fact that $$\min(A,B)\sim \mathcal E(\lambda_a+\lambda_b),$$ think about the first arrival time of the addition of two Poisson processes, for example.

From there, you could decompose the event $\{A&lt;B&lt;C\}$ into the intersection of the two independent events $$\{A&lt;\min(B,C)\}$$ and $$\{B&lt;C\}$$ to get an intuition of the result.
I have written a program to test the probabilities for three independent exponential random variables A, B and C with respective parameters λa = 1, λb = 2, λc = 3. Here are the results:

P (A &lt; B &lt; C) = 0.3275
P (A &lt; C &lt; B) = 0.2181
P (B &lt; A &lt; C) = 0.2047
P (B &lt; C &lt; A) = 0.0681
P (C &lt; A &lt; B) = 0.1211
P (C &lt; B &lt; A) = 0.0603


P (A &lt; B &lt; C) is not 1/15.

Here is the source code of the program:

Random Rand = new Random(2015);

public Double GetExponential(Double mean)
{
    return -mean * Math.Log(Rand.NextDouble());
}

var totalCase = 0;
for (int i = 0; i &lt; 1000000; ++i)
{
    var p1 = GetExponential(1);
    var p2 = GetExponential(2);
    var p3 = GetExponential(3);

    if (p1 &lt; p2 &amp;&amp; p2 &lt; p3)
        ++totalCase;
}
Console.WriteLine(totalCase / 1000000.0);