Answer to Question #320438 in Statistics and Probability for Klum

Question

in the new apartment, 2k new light bulbs are included in the lighting network. Each light bulb burns out within a year with probability p. Find the probability that less than half of the original light bulbs will have to be replaced with new ones within a year.

Answer 1

$P\left( \hat{p}<\frac{1}{2} \right) =P\left( \sqrt{n}\frac{\hat{p}-p}{\sqrt{p\left( 1-p \right)}}<\sqrt{n}\frac{\frac{1}{2}-p}{\sqrt{p\left( 1-p \right)}} \right) =\\=P\left( Z<\sqrt{2000}\frac{0.5-p}{\sqrt{p\left( 1-p \right)}} \right) =\varPhi \left( \sqrt{2000}\frac{0.5-p}{\sqrt{p\left( 1-p \right)}} \right)$