Answer:
for a) the time required is 228488.5 seconds= 63.469 hours
for b) the time required is 54 seconds
Explanation:
for a) since each combination is equally probable , then the number of possible combinations is
CT=Combinations = number of characters^length of password = 26⁴
then the number of combinations at time t will be the total , less the ones already tried:
Ct = CT - (n-1) , since n=α*t → Ct=CT-α*t
since each combination is equally probable , then the probability to succeed
pt = 1/Ct = 1/ (CT- α*t +1)
but the probability of having a success in time t , means also not succeeding in the previous trials , then
Pt = pt*П(1-pk), for k=1 to t-1
Pt = 1/ (CT- α*t +1) П[1-1/ (CT- α*k +1)] = 1/ (CT- α*t +1) П[(CT- α*k )] /(CT- α*k +1)]
since α=1 ,
Pt = 1/ (CT- t +1) П[(CT- k )] /(CT- k +1)] = 1/ (CT- t +1) * [CT- (t-1) ]/CT = 1/CT
then the expected value of the random variable t= time to discover the correct password is
E(t) = ∑ t* Pt = ∑ t *1/CT , for t=1 until t=CT/α =CT
E(t) = ∑ t *(1/CT) = (1/CT) ∑ t = (1/CT) * CT*(CT+1)/2 = (CT+1)/2
therefore
E(t) = (CT+1)/2 = (26⁴
+1)/2 = 228488.5 seconds = 63.469 hours
for b)
time required = time to find character 1 + time to find character 2 +time to find character 3 +time to find character 4 = 4*time to find character
since the time to find a character is the same case as before but with CT2=Combinations = 26 ,then
t= 4*tc
E(t) = 4*E(tc) = 4*(CT2+1)/2 = 4*(26+1)/2 = 54 seconds
