Get the solution to "Ali an Sara each choose a number independently..."

Cem Hayes

Answered question

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 [A \cap B]

Answer & Explanation

rogreenhoxa8

Skilled2021-09-14Added 109 answers

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

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