A letter is takenout at random from 'ASSISTANT' and another is taken out from 'STATISTICS'. The probability that they are the same letter is :

$ASSISTANT\to AAINSSSTT$

$STATISTICS\to ACIISSSTTT$

Here N and C are not common and same letters can be A, I, S, T. Therefore

 Probability of choosing A =   $\frac{2}{C}\times \frac{1}{C}$ = 1/45 

 Probability of choosing I =  $\frac{1}{9}\times \frac{2}{C}$ = 1/45

Probability of choosing S =  $\frac{3}{C}\times \frac{3}{C}$ = 1/10

Probability of choosing T =  $\frac{2}{C}\times \frac{3}{C}$ = 1/15

Hence, Required probability =    $\frac{1}{45}+\frac{1}{45}+\frac{1}{10}+\frac{1}{15}= \frac{19}{90}$