Why does the spiral of a positron have a larger radius than that of an electron in this picture in a bubble chamber? The smaller spiral is caused by an electron
The bigger spiral is caused by a positron
However, they have the same mass and magnitude of charge. So, during this pair production, why does the positron follow a spiral that has a larger radius?

 A: The electron and positron are the materialization of the energy of a gamma/photon hitting (let us assume)  a proton in a hydrogen bubble chamber. The pair cannot be created without an interaction in the field of another particle, due to four momentum conservation in the center of mass system. The photon has zero mass, the electron positron pair are limited by the mass of each particle : the center of mass system of the e+ e- the four vector invariant mass is at least  m_e+ + m_e-.
Energy and momentum balance will give the original energy of the photon. As a three body problem   the three particles involved can share the momentum within the constraints of the conservation laws. Nature needs no calculator.
A: In a magnetic field $\boldsymbol B$, a particle with charge $q$ moves in circles of radius $$r=\frac{m\,v}{|q|\,\|\boldsymbol B\|},\tag{1}$$ where $v$ is its speed. The orientation (clockwise or anticlockwise) depends on sign of $q$. Since electrons and positrons have the same masses and opposite charges, an electron and a positron in a magnetic field move in circles with opposite orientations. If they have the same speed they will move in circles of the same sizes because formula (1) gives the same radius. But if they have not the same speed, the circles will have different radii, accordingly.
As @annav said, when the electron and the positron are created, their momenta $p=mv$ have no reason to be equal, and one can observe either an electron with a larger circle than the positron or the opposite as in you picture. The case where they have exactly the same speed is highly improbable.
A: This is a so-called 'trident' event: a gamma converts to an electron-positron pair in the field of an electron in the target liquid. The electron gets a big enough kick to produce a track of its own, which is why there are two tracks curving right and one (the positron) curving left.
The way the photon energy is shared between the 3 particles is random.  It is not shared equally. It just happens that in this case the positron got more energy (hence more momentum, hence larger radius) than the low-energy electron. 
Likewise lower down there is a separate gamma conversion, this time the more usual case of a conversion in the field of a nucleus, which does not leave a track, and the electron and positron have different energies/momenta/curvature.
