perplexus.info :: Probability : Hexadecimal Hinder

Hexadecimal Hinder (Posted on 2016-06-19)

x is a randomly chosen hexadecimal real number on the interval (0, (10)₁₆)

Determine the probability of each of the following:

(i) x and 3^x have the same first digit.

(ii) x and x³ have the same first digit.

(iii) x³ and 3^x have the same first digit.

(iv) x, x³ and 3^x have the same first digit.

*** “First digit” denotes the first nonzero digit of the number when expressed in hexadecimal form.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

re: part (i) solution --- correction

Comment 6 of 6 |

(In reply to part (i) solution by Charlie)

In the solution to part (i) I said that the digit 1 occupies 1/16 of the fractional unit immediately after the hexadecimal point, and that this was supplemented by a portion of the zero segment, as 1/16. But this neglected double zero, etc.

The true complete fraction should be 1/15, as would be the case if we continued that infinite series: 1/16 + 1/16^2 + 1/16^3 + ....

That changes the first addent in the totals at the bottom, which should be:

.066666666666667

.261859507142914

.14031399558998

.0959032742893844

.07285801232989481

.058745493567895

making the overall total .6963469495867353, which, when divided by 16 becomes the probability .04352168434917095.

The discrepancy is small enough that the simulation would not give it away as obviously wrong.

Posted by Charlie on 2016-06-20 21:56:20

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