1.
leaving things to chance, trusting to luck
Relying on luck or fate rather than planning or effort. Often carries a slightly negative nuance, implying a lack of proper preparation.
運任せ means relying on luck rather than planning or effort. Often carries a slightly negative nuance, suggesting carelessness or lack of preparation.
Etymology
Compound of 運 (luck, fortune) + 任せ (leaving to, entrusting). The 〜任せ pattern appears in other words like 人任せ (leaving things to others) and 成り行き任せ (leaving things to take their course).