x and y are two real numbers, each of which is uniformly randomly chosen in the interval (1,5).
Determine the probability of each of the following events:
(i) ⌈3*x⌉ + ⌈4*y⌉ = ⌈3*x + 4*y⌉
(ii) ⌈3*x⌉ - ⌈4*y⌉ = ⌈3*x - 4*y⌉
(iii) ⌈3*x⌉ * ⌈4*y⌉ = ⌈12*x*y⌉
⌈3*x⌉
(iv) --------- = ⌈(3*x)/(4*y)⌉
⌈4*y⌉
*** ⌈n⌉ denotes ceiling(n), that is, the lowest integer greater than or equal to n.
The LHS can only equal one of {1,2,3,4} in the interval given.
On the RHS the numerator is one of {3,...,15} and denominator one of {4,...,20} there are only 3 cases.
Case 1: Both sides equal 3.
The RHS is between the lines y=x/4 and y=3x/8
The LHS requires the fraction 15/5 so ceil(3x)=15 and ceil(4y)=5
This region is a triangle bounded by y=5/4, x=14/3 and y=x/4 and has area 1/72
Case 2: Both sides equal 2.
The RHS is between the lines y=3x/8 and y=3x/4
The LHS requires the fraction be one of 14/7, 12/6, 10/5 so there are three triangles each with area 1/48.
Case 3: Both sides equal 1.
The RHS is above y=3/4
The LHS requires equal numerator and denominator.
There are 11 triangles each with area 1/24.
Total area = 1/72 + 4/48 + 11/24 = 77/144
Probability = 77/2304 = 0.0334201389
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Posted by Jer
on 2022-12-08 11:11:36 |
part (iv)
