I am working through a problem like this:
- Alice and Bob each pick a number in Uniform([0, 2])
- Let A = The magnitude of the difference of the two numbers is greater than 1/3.
- Let D = Alice’s number is greater than 1/3
- What is $P(A \cap D)$
I am attaching the solution below.
But what I am confused about is where the 5/3 and 4/3 come from. In other words, can somebody help explain this diagram?
Representing by $X$ and $Y$ the numbers drawn by Alice and Bob respectively (under independence) the area that appears in your solution is the union of two rectangular triangles:
The first triagle, above the main diagonal, has vertices $(1/3,2/3), (1/3,2), (5/3,2)$. This is the region described by $$\{(x,y)\in[0,2]^2:\frac13\leq x\}\cap\{(x,y)\in[0,2]^2: x-y\leq -\frac13\}$$ Its are is $\frac12\Big(2-\frac23\Big)\Big(\frac53-\frac13\Big)=\frac89$.
The second triangle, below the diagonal, has vertices $(\tfrac13,0), (2,\frac53), (2,0)$. This region is described by $$\{(x,y)\in[0,2]^2: \frac13\leq x\}\cap \{(x,y)\in[0,2]^2: x-y\geq\frac13\}$$ The are of this region is $\frac12\Big(2-\frac13\Big)\Big(\frac53-0\Big)=\frac{25}{18}$