Question

Let A,B, and C are iid Uniform (0,1) random variables. Use Monte Carlo simulation to estimate the probability that the random quadratic equation Ax²+Bx+C=0 has real roots. Use 3700 as the initial seed and 10000 as the number of simulation runs.

Answer 1

The probability that a random quadratic equation with coefficients uniformly distributed between 0 and 1 has real roots is estimated with a Monte Carlo simulation using 10000 runs and a seed of 3700. The equation's discriminant is calculated for each simulated trio of A, B, and C, and instances where it is nonnegative are counted to determine the probability.

Step-by-step explanation:

To estimate the probability that a random quadratic equation Ax² + Bx + C = 0, where A, B, and C are independent identically distributed (aid) Uniform (0,1) random variables, has real roots, we use a Monte Carlo simulation. For a quadratic equation to have real roots, its discriminant (B² - 4AC) must be nonnegative. By generating random values for A, B, and C within the uniform distribution and calculating this discriminant, one can determine if each simulated equation has real roots. Running this simulation for a large number of trials—10000 in this case—and using a set seed of 3700 for reproducibility, allows for the approximation of the probability.

Using a pseudo-random number generator with the specified seed, one can simulate the values of A, B, and C 10000 times, calculate the discriminant for each set, and then count the number of times the discriminant is greater than or equal to zero. Dividing this count by the total number of simulations gives an estimation of the probability that a random quadratic equation has real roots.