We propose a numerical method for quick evaluation of the probability that an asteroid will collide with a planet. The method is based on linear mappings of an expected moment of a close approach of the asteroid to the planet and the detection of collisions of the virtual objects with the massive body. The standard way for solving the problem of estimating the collision probability consists in simulating the evolution of the uncertainty cloud numerically based on the stepwise integration of virtual orbits. This is naturally associated with huge processor time costs. The proposed method is tested using the examples of the 2011 AG5 and 2007 VK184 asteroids that are presently in the top of the list of the most dangerous celestial objects. The test results show that linear mappings allow one to obtain the estimates of probabilities quicker by several orders than numerical integration of all virtual orbits.