When an elastic tube reinforced with helical fibres is inflated, its ends rotate. In large deformations, the amount and chirality of rotation is highly non-trivial, as it depends on the choice of strain–energy density and the arrangements of the fibres. For anisotropic hyperelastic tubes where the material parameters are single-valued constants, the problem has been satisfactorily addressed. However, in many systems, the material parameters are not precisely known, and it is therefore more appropriate to treat them as random variables. The problem is then to understand chirality in a probabilistic framework. Here, we develop a procedure for examining the elastic responses of a hyperelastic cylindrical tube of stochastic anisotropic material, where the material parameters are spatially-independent random variables defined by probability density functions. The tube is subjected to uniform dead loading consisting of internal pressure, axial tension and torque. Assuming that the tube wall is thin and that the resulting deformation is the combined inflation, extension and torsion from the reference circular cylindrical configuration to a deformed circular cylindrical state, we derive the probabilities of radial expansion or contraction, and of right-handed or left-handed torsion. We refer to these stochastic behaviours as ‘likely inflation’ and ‘likely chirality’, respectively. •Hyperelastic cylindrical tubes of stochastic anisotropic material are treated.•The material parameters are random variables defined by probability densities.•The deformation is the combined inflation, extension and torsion.•The probability of right-handed or left-handed torsion is characterised.•The probability of radial expansion or contraction is also obtained.