– 12 – model to design a code that would allow us to quantify this and to test a wide range of input assumptions. We conducted 13 simulations. Each was characterized by (a) the value of α, (b) the value of τ$_{monitor}$, (c) the range of masses of stars monitored by Kepler, (d) the rule used to compute the final core mass for systems experiencing stable mass transfer, (e) the ages of the stars monitored by Kepler. In this way, we were able to take into consideration the uncertainties in the input physics and to identify those results that are robust. The key question is: what fraction of all monitored systems are binaries in which a white dwarf is expected to transit a main-sequence star? To compute the number of monitored systems we counted members of the following sets of stars, considering only those in the appropriate range of masses: (1) all isolated stars; (2) stars that are the results of mergers; (3) stars with dwarf stellar companions having luminosity less than 0 . 001 their own luminosity $^{4}$; (4) stars in binaries so wide that the individual stars could be resolvable $^{5}$; and (5) stars orbited by white dwarfs. This last category includes all white-dwarf/main-sequence pairs, regardless of the formation history of the white dwarf. Figure 3 shows the result of one such simulation. In this particular simulation we generated 4 million stellar systems, and used α = 10 , and τ$_{monitor}$ = 1 year. We assumed that Kepler is monitoring main-sequence stars with masses in the range 0 . 9 − 2 . 9 M ⊙ . We assumed that, for systems undergoing stable mass transfer, the final value of the core mass would lie within 10% of its initial value ( δ < 0 . 1 M$_{c}$ ). There were approximately 370 , 000 stars with masses within the selected range that would appear to be single stars. We then identified all binaries in which the primary star fills its Roche lobe. Those with q > 1 . 5 were assumed to undergo common envelope evolution. For smaller values of q , we assumed stable mass transfer and selected the final value of the primary’s core mass. The numbers of binaries with the present-day primary having a mass in the monitored range and (1) surviving the common envelope, were ∼ 16 , 000; (2) having ended stable mass transfer, were ∼ 35 , 000 . Most of these will not produce transits. Taking the transit probability into account, we find that 3 . 6 × 10 − 3 of the monitored stars should exhibit at least one transit per year, in which the transiting object is a white dwarf that survived a common envelope. Many of these will exhibit multiple transits per year. The comparable fraction of systems with with transiting $^{4}$Depending on the orbital inclination these might provide transits that appear to be planetary. They could also exhibit deep eclipses like those seen in KOI-74 and KOI-81. $^{5}$Because we do not attempt to generate the 3-D spatial distribution of stars, it is not clear which binaries will actually be resolvable. We assumed that all binaries with orbital separation greater than 10 3 . 5 AU were potentially resolvable. By adding to the number of apparently isolated stars, we are lowering the fraction of systems with white dwarfs orbiting main sequence stars. This produces a more conservative estimate.