(ii)
The probability should be the same within any span of numbers starting with 1 and ending with 10 (i.e., 10 hex), such as 1 - 10 or .1 to 1 or .01 to .1, etc., as the digits of the cubes depend only on the digits of x.
So the probability within (0,10 hex) is the same as within (1,10 hex), a span of 15 (in decimal).
1 1
2 8
3 1B
4 40
5 7D
6 D8
7 157
8 200
9 2D9
A 3E8
B 533
C 6C0
D 895
E AB8
F D2F
10 1000
In the first range, 1 through cuberoot(2) qualify, so
2^(1/3) - 1 = .25992104989487, which is over a quarter of a unit on the number line,
which becomes the first addend in the sum that will be divided by 15.
When 2 plus a fraction is cubed, the result begins with 8 through F, or 1 -- never a 2.
When 3 plus a fraction is cubed, the result begins with 1 through 3:
(4*16)^(1/3) - (3*16)^(1/3) = .36575881433572, the second addend
When 4 plus a fraction is cubed, the result begins with 4 through 7:
(5*16)^(1/3) - (4*16)^(1/3) = .30886938006377, the third addend
5 plus a fraction results in from 7 through D-- nothing to count here.
In fact nothing else has matches until F:
(16^3)^(1/3) - (15*16^2)^(1/3) = .3405294353246, the last addend.
.25992104989487
.36575881433572
.30886938006377
.3405294353246
totalling 1.27507867961896. When divided by 15, it gives us
.08500524530793067 as the probability.
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Posted by Charlie
on 2016-06-19 18:44:57 |
part (ii) solution
