Brief.
Finding large prime numbers by Rabin-Miller test. After 5 tests the Theorem guarantees the probability to get a false result is less than (actually far less than) 1/1024. The correctness can be easily significantly improved by increasing number of tests.
Caculate the base of natural logarithm in decimal notation. The Theorem about Taylor series with Remainder is applied to obtain an upper bound of the error.
Demenstrate Newton's law of gravitation and that planet orbits are one of the three quadratic curves. Hit P to generate a planet.
Conway's Game of Life. A live cell lives iff it has 2 or 3 live neighbors. A live cell generates iff it has exactly 3 neighbors.