let-x-and-y-be-random-numbers-from-0-to-10-where-x-y-R-x-y-d-what-is-the-probability-that-their-sum-is-less-than-10-in-the-following-cases-a-d-0-b-d-1-c-d-2-

Question Number 205394 by Red1ight last updated on 19/Mar/24

let x and y be random numbers from 0 to 10

where x, y \in R

∣ x - y ∣⩾ d

what is the probability that their sum is less than 10

in the following cases

a) d = 0

b) d = 1

c) d = 2

Answered by mr W last updated on 20/Mar/24

Commented by mr W last updated on 20/Mar/24

∣ x - y ∣⩾ d

\Rightarrow {\begin{cases} x - y ⩾ d \Rightarrow y ⩽ x - d \\ o r \\ x - y ⩽ - d \Rightarrow y ⩾ x + d \end{cases}

∣ x - y ∣⩽ d m e a n s w h e n p o i n t l i e s i n

h a t c h e d z o n e s .

x + y ⩽ 10 \Rightarrow y ⩽ 10 - x \Rightarrow s h a d e d a r e a

p = \frac{{(10 - d)}^{2}}{2 \times 10^{2}} = \frac{1}{2} {(1 - \frac{d}{10})}^{2}

p_{d = 0} = 0.5

p_{d = 1} = 0.405

p_{d = 2} = 0.32

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