Answer:
The smallest number of servers required is 3 servers
Explanation:
Given
[tex]Reliability = 99.98\%[/tex]
[tex]Individual Servers = 95\%[/tex]
Required
Minimum number of servers needed
Let p represent the probability that a server is reliable and the probability that it wont be reliable be represented with q
Such that
[tex]p = 95\%\\[/tex]
It should be noted that probabilities always add up to 1;
So,
[tex]p + q = 1[/tex]'
Subtract p from both sides
[tex]p - p + q = 1 - p[/tex]
[tex]q = 1 - p[/tex]
Substitute [tex]p = 95\%\\[/tex]
[tex]q = 1 - 95\%[/tex]
Convert % to fraction
[tex]q = 1 - \frac{95}{100}[/tex]
Convert fraction to decimal
[tex]q = 1 - 0.95[/tex]
[tex]q = 0.05[/tex]
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To get an expression for one server
The probabilities of 1 servers having 99.98% reliability is as follows;
[tex]p = 99.98\%[/tex]
Recall that probabilities always add up to 1;
So,
[tex]p + q = 1[/tex]
Subtract q from both sides
[tex]p + q - q = 1 - q[/tex]
[tex]p = 1 - q[/tex]
So,
[tex]p = 1 - q = 99.98\%[/tex]
[tex]1 - q = 99.98\%[/tex]
Let the number of servers be represented with n
The above expression becomes
[tex]1 - q^n = 99.98\%[/tex]
Convert percent to fraction
[tex]1 - q^n = \frac{9998}{10000}[/tex]
Convert fraction to decimal
[tex]1 - q^n = 0.9998[/tex]
Add [tex]q^n[/tex] to both sides
[tex]1 - q^n + q^n= 0.9998 + q^n[/tex]
[tex]1 = 0.9998 + q^n[/tex]
Subtract 0.9998 from both sides
[tex]1 - 0.9998 = 0.9998 - 0.9998 + q^n[/tex]
[tex]1 - 0.9998 = q^n[/tex]
[tex]0.0002 = q^n[/tex]
Recall that [tex]q = 0.05[/tex]
So, the expression becomes
[tex]0.0002 = 0.05^n[/tex]
Take Log of both sides
[tex]Log(0.0002) = Log(0.05^n)[/tex]
From laws of logarithm [tex]Loga^b = bLoga[/tex]
So,
[tex]Log(0.0002) = Log(0.05^n)[/tex] becomes
[tex]Log(0.0002) = nLog(0.05)[/tex]
Divide both sides by [tex]Log0.05[/tex]
[tex]\frac{Log(0.0002)}{Log(0.05)} = \frac{nLog(0.05)}{Log(0.05)}[/tex]
[tex]\frac{Log(0.0002)}{Log(0.05)} = n[/tex]
[tex]n = \frac{Log(0.0002)}{Log(0.05)}[/tex]
[tex]n = \frac{-3.69897000434}{-1.30102999566}[/tex]
[tex]n = 2.84310893421[/tex]
[tex]n = 3 (Approximated)[/tex]
Hence, the smallest number of servers required is 3 servers
