Karnataka Board PUCPUC Science 2nd PUC Class 12

Find the Mean and Variance of the Number of Tails in Three Tosses of a Coin. - Mathematics

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Sum

Find the mean and variance of the number of tails in three tosses of a coin.

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Solution

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
\[\frac{1}{8}\]
1
\[\frac{3}{8}\]
2
\[\frac{3}{8}\]
3
\[\frac{1}{8}\]

Computation of mean and variance

xi pi pixi pixi2
0
\[\frac{1}{8}\]
0 0
1
\[\frac{3}{8}\]
\[\frac{3}{8}\]
\[\frac{3}{8}\]
2
\[\frac{3}{8}\]
\[\frac{6}{8}\]
\[\frac{12}{8}\]
3
\[\frac{1}{8}\]
\[\frac{3}{8}\]
\[\frac{9}{8}\]
    `∑`pixi = \[\frac{3}{2}\]
 
`∑`pixi2=3

\[\text{ Mean } = \sum p_i x_i = \frac{3}{2}\]
\[\text{ Variance } = \sum p_i {x_i}^2 - \left( \text{ Mean}  \right)^2 \]
\[ = 3 - \left( \frac{3}{2} \right)^2 \]
\[ = 3 - \frac{9}{4}\]
\[ = \frac{3}{4}\]

  Is there an error in this question or solution?
Chapter 32: Mean and Variance of a Random Variable - Exercise 32.2 [Page 43]

APPEARS IN

RD Sharma Class 12 Maths
Chapter 32 Mean and Variance of a Random Variable
Exercise 32.2 | Q 4 | Page 43

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