Exercise 1

Let S100 be the number of heads that turn up in 100 tosses of a fair coin. Use the Central Limit Theorem to estimate (a) P(S100 ≤ 45).

n = 100
p = 0.5
q = 1-p
sd = sqrt(n*p*q)
expected = n*p

value = (45.5 - expected) / sd
pnorm(value)
## [1] 0.1840601
  1. P(45 < S100 < 55).
n = 100
p = 0.5
q = 1-p
sd = sqrt(n*p*q)
expected = n*p

value1 = (44.5 - expected) / sd
value2 = (54.5 - expected) / sd
pnorm(value2) - pnorm(value1)
## [1] 0.6802738
  1. P(S100 > 63).
n = 100
p = 0.5
q = 1-p
sd = sqrt(n*p*q)
expected = n*p

value1 = (63.5 - expected) / sd

pnorm(value1,lower.tail=FALSE)
## [1] 0.003466974
  1. P(S100 < 57).
n = 100
p = 0.5
q = 1-p
sd = sqrt(n*p*q)
expected = n*p

value1 = (56.5 - expected) / sd

pnorm(value1)
## [1] 0.9031995