$($Corrected$)$ Let $A$ be defined as

$A=\{win\{a,b{\}}^{**}|w$ has two $a\text{'}s$ separated $byan$ odd number $of$ symbols $\}$

For example, babbaabbbabbbba and abababba are strings in $A,$ but bbabbb, bbabb and bbabbab

are not.

Note: every xin$\{a,b{\}}^{**}$ that has $3$ a$\text{'}$s must be in $A,$ because if there are an odd number of symbols

between the first a and the second $a$ or an odd number of symbols between the second a and the

third a then xinA. On the other hand, if there's an even number of symbols between the first and

second a and between the second and third a$,$ then there's an odd number of symbols between the

first and third a $($since even $+1+$ even is odd$).$

Draw the smallest NFA you can for $A.$ Then, convert your NFA to an equivalent DFA using the

"subset construction" method $($Theorem $1\mathrm{.}39).$