Many studies investigate the decisions made by animals by focussing on their individual attitudes towards risk, i.e., risk seeking, risk neutrality or risk aversion. However, little attention has been paid to how far individuals understand the different odds of outcomes. In a previous gambling task involving up to 18 different lotteries (Pelé et al., 2014), non-human primates used probabilities of gains and losses to make their decision. Although the use of complex mathematical calculation for decision-making seemed unlikely, we applied a gradual decrease in the chances to win throughout the experiment. This probably facilitated the extraction of information about odds. Here, we investigated whether individuals would still make efficient decisions if this facilitating factor was removed. To do so, we randomized the order of presentation of the 18 lotteries. Individuals from four ape and two monkey species were tested. Only capuchin monkeys differed in their gambling behaviour, playing even when there was nothing to win. Randomising the lottery presentation order leads all species to predominantly use a maximax heuristic in which individuals gamble as soon as there is at least one chance to win more than they already possess, whatever the risk. Most species also gambled more as the frequency of larger rewards increased. These results suggest the occurrence of optimistic behaviour. The maximax heuristic is sometimes observed in human managerial and financial decision-making, where risk is ignored for potential gains, however low they may be. Our results suggest a shared and strong propensity in primates to rely on heuristics whenever complexity in evaluation of outcome odds arises.