The outcome distribution of a sum of n identical monetary lotteries (Sn=L1+L2+…+Ln) is described with a Markov model. A decision maker with the alternation bias believes in more negative autocorrelation between lotteries and perceives Sn as a less risky asset (lower variance) than a rational agent does. Also the expected utility of Sn for a risk averse (risk seeking) individual is higher (lower) if she is a believer in the alternation bias. This theoretical result can be applied to the analysis of decisions on repeated investments and turns to be a plausible explanation for Samuelson's fallacy of large numbers.