Not a paradox...here is a proof that it is 1&#x2F;3<p>Let P(T) = probability that coin landed on tails, P(H) = probability that coin landed on heads.<p>Let &quot;1st&quot; denote the event that it is the first time you are awaken, &quot;2nd&quot; the second time.<p>Note that P(T|1st) = 1&#x2F;2 (if you were told that this is the first time you were woken up, it&#x27;s equally likely that the coin landed heads or tails). And of course, P(T|2nd) = 1<p>By the definition of conditional probability, 
1&#x2F;2 = P(T|1st) = P(1st|T)<i>P(T)&#x2F;P(1st) = (1&#x2F;2)</i>P(T)&#x2F;P(1st)
Hence P(T) = P(1st) = P(T)<i>1&#x2F;2 + P(H) = P(T)</i>1&#x2F;2+(1-P(T)) = 1 - P(T)&#x2F;2
-&gt; P(T) = 2&#x2F;3