Question

# Ali an Sara each choose a number independently and uniformly at random from interval [0,2]. Consider the following events:A:The absolute difference between the two numbers is greater than 1/4B: Alis number is greater than 1/4 Find the probability P[A\cap B]

Probability
Ali an Sara each choose a number independently and uniformly at random from interval [0,2].
Consider the following events:
A:The absolute difference between the two numbers is greater than 1/4
B: Alis number is greater than 1/4
Find the probability $$\displaystyle{P}{\left[{A}\cap{B}\right]}$$

2021-09-14

We observe that $$|x-y|>\frac{1}{4}$$
$$<=>x-y>\frac{1}{4}$$ or $$x-y<-\frac{1}{4}$$.
$$\displaystyle{P}{\left({A}\cap{B}\right)}={\frac{{\text{area of the sgaded region}}}{{\text{area of the square of side 2}}}}$$
Now, (I)is a right angled triangle with $$base =\frac{7}{4}$$, $$altitude=2-\frac{1}{4}=\frac{7}{4}$$
-> Hence area of (I) is $$\displaystyle{\frac{{\text{1}}}{{\text{2}}}}\cdot{\left({\frac{{\text{7}}}{{\text{4}}}}\right)}^{{2}}$$
(II)is also a right angled triangle with:$$\displaystyle{b}{a}{s}{e}{2}-{\frac{{\text{1}}}{{\text{2}}}}={\frac{{\text{3}}}{{\text{2}}}}{\quad\text{and}\quad}{a}{lt}{i}{t}{u}{d}{e}={\frac{{\text{7}}}{{\text{4}}}}-{\frac{{\text{1}}}{{\text{4}}}}={\frac{{\text{6}}}{{\text{4}}}}$$
Hence area of (II) is $$\displaystyle{\frac{{\text{1}}}{{\text{2}}}}\cdot{\frac{{\text{3}}}{{\text{2}}}}\cdot{\frac{{\text{6}}}{{\text{4}}}}={\left({\frac{{\text{3}}}{{\text{2}}}}\right)}^{{2}}\cdot{\frac{{\text{1}}}{{\text{2}}}}$$
$$\displaystyle\therefore{A}{r}{e}{a}\ {o}{f}\ {s}{h}{a}{d}{e}{d}\ {r}{e}{g}{i}{o}{n}={\frac{{\text{1}}}{{\text{2}}}}\cdot{\left({\left\lbrace{\frac{{\text{49}}}{{\text{16}}}}\right\rbrace}+{\frac{{\text{9}}}{{\text{4}}}}\right)}{s}{q}{y}{a}{r}{e}\ {u}{n}{i}{t}{s}={\frac{{\text{85}}}{{\text{32}}}} {u}{n}{i}{t}{s}.$$
$$\displaystyle{P}{\left({A}\cap{B}\right)}={\frac{{\text{85/32}}}{{\text{2*2}}}}={\frac{{\text{85}}}{{\text{128}}}}$$