Numerical probabilistic The answer is always an approximation. Prabhas Chongstitvatana

Prabhas Chongstitvatana The answer is always an approximation. The expected precision improves given more time. The expected precision improves given more time. Prabhas Chongstitvatana

Prabhas Chongstitvatana The answer is always an approximation. The expected precision improves given more time. The error is inverse proportion to the square root of the amount of work. (100 times more work to obtain one additional digit of precision) Example : use in simulation Prabhas Chongstitvatana

Prabhas Chongstitvatana Buffon’s needle 18th century, George Louis Leclerc, compte de Buffon. Probability that a needle will fall across a crack is 1/pi (each drop is independent to the others) Plank width = w Needle length L = w/2 Prabhas Chongstitvatana

Prabhas Chongstitvatana Approximate pi : n/k as an estimator of pi Approximate w : w >= L , w is estimated by Prabhas Chongstitvatana

Prabhas Chongstitvatana How fast this ‘algorithm’ converge? Convergence analysis Estimate pi : Xi each needle Xi =1 if i-th needle fall across a crack, 0 otherwise. Prabhas Chongstitvatana

Prabhas Chongstitvatana X estimate of 1/pi after dropping n needles. Prabhas Chongstitvatana

Prabhas Chongstitvatana X is normal distributed Prabhas Chongstitvatana

Prabhas Chongstitvatana X is normal distributed Prabhas Chongstitvatana

Prabhas Chongstitvatana We want to estimate pi not 1/pi when With n needles, estimate pi will have less precision than estimate 1/pi by one digit. Prabhas Chongstitvatana

Prabhas Chongstitvatana Given n , the value of pi is between and with probability at least p Prabhas Chongstitvatana

Prabhas Chongstitvatana Given n , the value of pi is between and Example : How many n to estimate pi within 0.01 of the correct value with the confidence 99% ? precision 0.001 (one more digit than estimate 1/pi) n >1.44 million Prabhas Chongstitvatana