The influence of weak shear thinning on the onset of chaos in thermal convection is examined for a Carreau-Bird fluid. A truncated Fourier representation of the flow and temperature fields leads to a three-dimensional system that generalizes the classical Lorenz system for a Newtonian fluid. It is found that the critical Rayleigh number at the onset of thermal convection remains the same as for a Newtonian fluid, but the amplitude and nature of the convective cellular structure is dramatically altered by shear thinning, The presence of shear thinning leads to a second Hopf bifurcation around the convective branches in addition to the one usually present in the Lorenz system. While chaotic behavior sets in, as the Rayleigh number increases, at the first Hopf bifurcation similarly to the case of a Newtonian fluid, there appears a series of periodic behaviors (inverse period doubling) leading to intermittency and again to chaos at a Rayleigh number that becomes increasingly smaller as the effect of shear thinning increases.