Cem Hayes

2021-09-13

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: Ali`s number is greater than 1/4
Find the probability $P\left[A\cap B\right]$

rogreenhoxa8

We observe that $|x-y|>\frac{1}{4}$
$<=>x-y>\frac{1}{4}$ or $x-y<-\frac{1}{4}$.
$P\left(A\cap B\right)=\frac{\text{area of the sgaded region}}{\text{area of the square of side 2}}$
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 $\frac{\text{1}}{\text{2}}\cdot {\left(\frac{\text{7}}{\text{4}}\right)}^{2}$
(II)is also a right angled triangle with:$base2-\frac{\text{1}}{\text{2}}=\frac{\text{3}}{\text{2}}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}altitude=\frac{\text{7}}{\text{4}}-\frac{\text{1}}{\text{4}}=\frac{\text{6}}{\text{4}}$
Hence area of (II) is $\frac{\text{1}}{\text{2}}\cdot \frac{\text{3}}{\text{2}}\cdot \frac{\text{6}}{\text{4}}={\left(\frac{\text{3}}{\text{2}}\right)}^{2}\cdot \frac{\text{1}}{\text{2}}$

$P\left(A\cap B\right)=\frac{\text{85/32}}{\text{2*2}}=\frac{\text{85}}{\text{128}}$

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