When a gravitationally lensed source crosses a caustic , a pair of images is created or destroyed .
We calculate the mean number of such pairs of micro-images \langle n \rangle for a given macro-image of a gravitationally lensed point source , due to microlensing by the stars of the lensing galaxy .
This quantity was calculated by Wambsganss , Witt & Schneider ( 1992 ) for the case of zero external shear , \gamma = 0 , at the location of the macro-image .
Since in realistic lens models a non-zero shear is expected to be induced by the lensing galaxy , we extend this calculation to a general value of \gamma .
We find a complex behavior of \langle n \rangle as a function of \gamma and the normalized surface mass density in stars \kappa _ { * } .
Specifically , we find that at high magnifications , where the average total magnification of the macro-image is \langle \mu \rangle = | ( 1 - \kappa _ { * } ) ^ { 2 } - \gamma ^ { 2 } | ^ { -1 } \gg 1 , \langle n \rangle becomes correspondingly large , and is proportional to \langle \mu \rangle .
The ratio \langle n \rangle / \langle \mu \rangle is largest near the line \gamma = 1 - \kappa _ { * } where the magnification \langle \mu \rangle becomes infinite , and its maximal value is 0.306 .
We compare our semi-analytic results for \langle n \rangle to the results of numerical simulations and find good agreement .
We find that the probability distribution for the number of extra micro-image pairs is reasonably described by a Poisson distribution with a mean value of \langle n \rangle , and that the width of the macro-image magnification distribution tends to be largest for \langle n \rangle \sim 1 .