A fair coin is flipped 101 times then a What number of heads

A fair coin is flipped 101 times, then:

a.) What number of heads is the most probable? (highest probability)

b.) What is the mean of the distribution?

c.) What is the variance of the distribution?

d.) What is the standard deviation for the distribution?

The coin is fair.

Hence if x is the no of heads then x is binomial with p = 0.5

n = 101

a) Binomial prob P(X=r) = nCr(0.5)ⁿ

nCr is the greatest if r = n/2 or (n+1)/2

(Recollect Pascal triangle where middle terms would be the highest.)

Hence when r = 51, probability is highest.

51 number of heads is the most probable

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b) Mean of the distribution = Mean of Binomial

= np = 101(0.5)

= 50.5

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c) Variance of the distribution = npq

= 51.5(0.5)

= 25.25

d) Std deviation of the distribution

= square root of variance

= 5.0249