Suppose one hundred numbers [math]X_1[/math], [math]X_2[/math], \dots, [math]X_{100}[/math] are chosen independently at random from [math][0,20][/math]. Let [math]S = X_1 + X_2 +\cdots+ X_{100}[/math] be the sum, [math]A = S/100[/math] the average, and [math]S^* = (S - 1000)/(10/\sqrt3)[/math] the standardized sum. Find lower bounds for the probabilities

• [math]P(|S - 1000| \leq 100)[/math].
• [math]P(|A - 10| \leq 1)[/math].
• [math]P(|S^*| \leq \sqrt3)[/math].