Let S100 be the number of heads that turn up in 100 tosses of a fair coin. Use the Central Limit Theorem to estimate

  1. P(S100 ≤ 45).

\[E(x) = np = 100 \cdot 0.5 = 50 \\ \sigma^2 = \sqrt{npq} = \sqrt{100 \cdot 0.5 \cdot 0.5} = \sqrt{25} = 5\]

\[\begin{align} P(S_{100} \leq 45) &= P(S_{n}^* \leq \frac{45 + .5 - 50}{5}) \\ &= P(S_{n}^* \leq \frac{45.5 - 50}{5}) \\ &= P(S_{n}^* \leq \frac{-4.5}{5}) \\ &= NA(-\infty,-0.9) \end{align}\]

pnorm(-0.9)
## [1] 0.1840601
  1. P(45 < S100 < 55).

\[\begin{align} P(45 < S_{100} < 55) &= P(\frac{45 - .5 - 50}{5} < S_{n}^* < \frac{55 +.5 - 50}{5}) \\ &= P(\frac{44.5 - 50}{5} < S_{n}^* < \frac{55.5 - 50}{5}) \\ &= P(\frac{-5.5}{5} < S_{n}^* < \frac{5.5}{5}) \\ &= NA(-1.1,1.1) \\ &= 2NA(0,1.1) \end{align}\]

pnorm(1.1) - pnorm(-1.1)
## [1] 0.7286679
  1. P(S100 > 63).

\[\begin{align} P(S_{100} > 63) &= 1 - P(S_{100} < 63) \\ &= 1 - P(S_{n}^* < \frac{63 + .5 - 50}{5}) \\ &= 1 - P(S_{n}^* < \frac{63.5 - 50}{5}) \\ &= 1 - P(S_{n}^* < \frac{13.5}{5}) \\ &= 1- NA(-\infty,2.7) \end{align}\]

1- pnorm(2.7)
## [1] 0.003466974
  1. P(S100 < 57).

\[\begin{align} P(P(S_{100} < 57) &= P(S_{n}^* < \frac{57 + \frac{1}{2} - 50}{5}) \\ &= P(S_{n}^* < \frac{57.5 - 50}{5}) \\ &= P(S_{n}^* < \frac{7.5}{5}) \\ &= NA(-\infty,1.5) \end{align}\]

pnorm(1.5)
## [1] 0.9331928