When asked to mentally simulate coin tosses, people generate sequences that differ systematically from those generated by fair coins. It has been rarely noted that this divergence is apparent already in the very 1st mental toss. Analysis of several existing data sets reveals that about 80% of respondents start their sequence with Heads. We attributed this to the linguistic convention describing coin toss outcomes as "Heads or Tails," not vice versa. However, our subsequent experiments found the "first-toss" bias reversible under minor changes in the experimental setup, such as mentioning Tails before Heads in the instructions. We offer a comprehensive account in terms of a novel response bias, which we call reachability. It is more general than the 1st-toss bias, and it reflects the relative ease of reaching 1 option compared to its alternative in any binary choice context. When faced with a choice between 2 options (e.g., Heads and Tails, when "tossing" mental coins), whichever of the 2 is presented first by the choice architecture (hence, is more reachable) will be favored. This bias has far-reaching implications extending well beyond the context of randomness cognition; in particular, to binary surveys (e.g., accept vs. reject) and tests (e.g., True--False). In binary choice, there is an advantage to what presents first.