# A letter is known to have come either from TATA NAGAR or from CALCUTTA.On the envelope,just two consecutive letter TA are visible.What is the probability that the letter came from TATA NAGAR.

Toolbox:
• $$E_1=letter \;is\;from\;T\;A\:T\:A\:N\;A\:G\:A\;R$$
• $$E_2=ietter\;is\;from\;C\;A\;L\;C\;U\;T\;T\;A$$
• $$A=event\;(T\;A)\;appers\;together$$
• $$E_1/A=P(letter \;is\;from\;T\;A\:T\:A\:N\;A\:G\:A\;R\;(T\;A)\;appers\;together$$)
• =$$\Large\frac{{P(E_1)}{P(A/E_1)}}{{P(E_1)}{P(A/E_1)}+{P(E_2)}{P(A/E_2)}}$$
• $$P(E_1)=P(E_2)\frac{1}{2}[each\;have\;equal\;chance$$]
$$there\;are\;eight\;possible\;ways\;of\;two\;letters\;apperaring\;from\;T\;A\;T\;A\;N\;A\;G\;A\;R$$
{$$(T\;A)(A\;T)(T\;A)(A\;N)(N\;A)(A\;G)(G\;A)(A\;R)$$}
$$P(A/E_1$$)=$$\Large\frac{2}{8}$$ =2 ways of getting (T"A)together
$$there\;are\;eight\;possible\;ways\;of\;two\;letters\;apperaring\;from\;C\;A\;L\;C\;U\;T\;T\;A$$
{$$(C\;A)(A\;L)(L\;C)(C\;U)(U\;T)(T\;T)(T\;A)$$}
=$$1\; ways\; of \;getting\; (T"A)\;together$$
$$P(A/E_2$$)=$$\Large\frac{1}{7}$$
P($$E_1/A)=P(letter \;is\;from\;T\;A\:T\:A\:N\;A\:G\:A\;R\;(T\;A)\;appers$$)
$$\Large\frac{\frac{1}{2}\times\frac{2}{8}}{\frac{1}{2}\times\frac{2}{8}+\frac{1}{2}\times\frac{1}{7}}$$
$$\Large\frac{\frac{1}{4}}{\frac{11}{28}}$$

=$$\Large\frac{7}{11}$$

edited Jun 4, 2013