Question

4. A fair two-sided coin is tossed repeatedly. (a) Find the expected number of tails until the first head is flipped. (b) Find the probability that there are exactly 5 heads in the first 10 flips. (c) Use the central limit theorem/normal approximation to approximate the probability that in the first 100 flips, between 45 and 55 of the flips are heads.

Answer 1

a)

here probability of a head for a fair coin p=1/2

therefore expected number of tails untill first head =1/p-1=1/(1/2)-1=1

b)

P(5 heads in first 10 flips)=¹⁰C₅(1/2)⁵(1/2)⁵ =0.2461

c)

n=	100	p=	0.5000
here mean of distribution=μ=np=		50
and standard deviation σ=sqrt(np(1-p))=		5.0000

therefore from normal approximation of binomial distribution:

probability =

P(45<X<55)

=

P(-1<Z<1)=

0.8413-0.1587=

0.6826