Journal article: A memoryless agent can acquire arbitrarily many signals. After each signal observation, she either terminates and chooses an action, or she discards her observation and draws a new signal. By conditioning the probability of termination on the information collected, she controls the correlation between the payoff state and her terminal action. We provide an optimality condition for the emerging stochastic choice. The condition highlights the benefits of selective memory applied to the extracted signals. Implications—obtained in simple examples—include (i) confirmation bias, (ii) speed-accuracy complementarity, (iii) overweighting of rare events, and (iv) salience effect.