Find the mean, variance and standard deviation of the number of tails in two tosses of a coin

In two tosses of a coin, the sample space is given byS = {HH, HT, TH, TT}

\(\therefore\) n(S) = 4

So, every single outcome has a probability \(\frac{1}{4}\)Let X= number of tails in two tosses.In two tosses, we may have no tail, 1 tail or 2 tails.

So, the possible values of X are 0,1,2 P(X = 0) = P(getting no tail) = P(H H) = \(\frac{1}{4}\)

P(X = 1) = P(getting } 1 tail) = P(HT \text { or } T H)= \(\frac{2}{4}=\frac{1}{2}\)
P(X = 2) = P(getting } 2 tails) = P(TT) = \(\frac{1}{4}\)Hence, the probability distribution of X is given by

X = xi	0	1	2
pi	\(\frac{1}{4}\)	\(\frac{1}{2}\)	\(\frac{1}{4}\)

\(\therefore\) mean, \(\mu=\Sigma x_{i} p_{i}=\left(0 \times \frac{1}{4}\right)+\left(1 \times \frac{1}{2}\right)+\left(2 \times \frac{1}{4}\right)\) = 1
Variance, \(\sigma^{2}=\Sigma x_{i}^{2} p_{i}-\mu^{2}\)
\(=\left[\left(0 \times \frac{1}{4}\right)+\left(1 \times \frac{1}{2}\right)+\left(4 \times \frac{1}{4}\right)\right]-1^{2}\)
= \(\frac{1}{4}\)
Standard deviation, \(\sigma=\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2}=\frac{1.414}{2}\) = 0.707
MEAN =1, VARIANCE= \(\frac{1}{4}\) = 0.25, STANDARD DEVIATION= 0.707

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Let X denote the number of tails in three tosses of a coin. Then, X can take the values 0, 1, 2 and 3.
Now,

\[P\left( X = 0 \right) = P\left( HHH \right) = \frac{1}{8}, P\left( X = 1 \right) = P\left( \text{ THH or HHT or HTH }\right) = \frac{3}{8}\]
\[P\left( X = 2 \right) = P\left( \text{ TTH or THT or HTT }\right) = \frac{3}{8}, P\left( X = 3 \right) = P\left( TTT \right) = \frac{1}{8}\]

Thus, the probability distribution of X is given by

x	P(X)
0
1
2
3

Computation of mean and step deviation

xi	pi	pixi	pixi2
0		0	0
1
2
3
		`∑`pixi =\[\frac{3}{2}\]	`∑` pixi2=3

\[\text{ Mean} = \sum p_i x_i = \frac{3}{2}\]\[\text{ Variance } = \sum p_i {x_i}2^{}_{} - \left( \text{ Mean } \right)^2 \]\[ = 3 - \left( \frac{3}{2} \right)^2 \]\[ = \frac{3}{4}\]\[\text{ Step Deviation} = \sqrt{\text{ Variance} }\]\[ = \sqrt{\frac{3}{4}}\]

\[ = 0 . 87\]