Question

Determine variance and standard deviation of the number of heads in three tosses of a coin.

Answer 1

Let X denote the number of heads tossed.

So, X can take the values 0, 1, 2, 3.

When a coin is tossed three times, we get

Sample space S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

P(X = 0) = P(no head) = P(TTT) = `1/8`

P(X = 1) = P(one head) = P(HTT, THT, TTH) = `3/8`

P(X = 2) = P(two heads) = P(HHT, HTH, THH) = `3/8`

P(X = 3) = P(three heads) = P(HHH) = `1/8`

Thus the probability distribution of X is:

X	0	1	2	3
P(X)	`1/8`	`3/8`	`3/8`	`1/8`

Variance of X = `sigma^2 = sumx_"i"^2"p"_"i" - mu^2`  .....(1)

Where µ is the mean of X given by

`mu = sumx_"i""p"_"i" = 0 xx 1/8 + 1 xx 3/8 + 2 xx 3/8 + 3 xx 1/8`

= `3/2`   .......(2)

Now `sumx_"i"^2"p"_"i" = 0^2 xx 1/8 + 1^2 xx 3/8 + 2^2 xx 3/8 + 2^2 xx 3/8 + 3^2 xx 1/8` = 3  ......(3)

From (1), (2) and (3), we get

`sigma^2 = 3 - (3/2)^2 = 3/4`

Standard deviation = `sqrt(sigma^2)`

= `sqrt(3/4)`

= `sqrt(3)/4`.