Flip a coin 100 times. We want the chance of getting exactly 50 heads. What is the exact probability, correct to six decimal places? What is the Normal approximation of the probability, to six decimal places?

Answer: a) 0.079589 b) 0.079656Step-by-step explanation:Since we have given that Number of times a coin is flipped = 100 timesNumber of times he get exactly head = 50Probability of getting head = [tex]\dfrac{1}{2}[/tex]We will use "Binomial distribution":Probability would be [tex]^{100}C_{50}(\dfrac{1}{2})^{50}(\dfrac{1}{2})^50\\\\=0.079589[/tex]Using "Normal approximation":n = 100p = 0.5So, mean = [tex]np=100\times 0.5=50[/tex]Standard deviation is given by[tex]\sqrt{np(1-p)}\\\\=\sqrt{50(1.05)}\\\\=\sqrt{50\times 0.5}\\\\=\sqrt{25}\\\\=5[/tex]So,[tex]P(X<x)=P(Z<\dfrac{\bar{x}-\mu}{\sigma})\\\\So, P(X=50)=P(49.5<X<50.5)\\\\=P(\dfrac{49.5-50}{5}<Z<\dfrac{50.5-50}{5})\\\\=P(-0.1<Z<0.1)\\\\=P(Z<0.1)-P(Z<-0.1)\\\\=0.539828-0.460172\\\\=0.079656[/tex]Hence, a) 0.079589 b) 0.079656