91Ó°ÊÓ

a.

A sample point refers to a particular value of any variable taken into consideration. In this context, a sample point refers to the observed sides of a coin when the three coins are tossed while doing an experiment.

There are 8 possible outcomes in the table above when the three coins are tossed where H and T represent head and tail, respectively.As x is considered to be H, the assigned values represent the number of heads that can occur when the three coins are tossed.

A fair coin has two sides and is denoted as head and tail.The probability of the occurrence of a side remains equal to half as both of them have an equal chance of occuring when it is tossed once.

The calculation of the probability of occurrence of Heads is shown below:

$\begin{array}{rcl}\mathbf{\text{P}}\left(\text{0}\right)\mathbf{\text{:P}}\left(\text{Zero} \text{heads}\right)& \mathbf{=}& \frac{\mathbf{\text{Favorable}}\mathbf{ }\mathbf{\text{events}}\mathbf{ }\mathbf{\text{showing}}\mathbf{ }\mathbf{\text{0}}\mathbf{ }\mathbf{\text{heads}}}{\mathbf{\text{Total}}\mathbf{ }\mathbf{\text{events}}\mathbf{ }}\\ \mathbf{\text{P}}\left(\text{1}\right)\mathbf{\text{:P}}\left(\text{One} \text{head}\right)& \mathbf{=}& \frac{\mathbf{\text{Favorable}}\mathbf{ }\mathbf{\text{events}}\mathbf{ }\mathbf{\text{showing}}\mathbf{ }\mathbf{\text{1}}\mathbf{ }\mathbf{\text{head}}}{\mathbf{\text{Total}}\mathbf{ }\mathbf{\text{events}}\mathbf{ }}\\ & \mathbf{=}& \frac{\mathbf{1}}{\mathbf{8}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{8}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{8}}\\ & \mathbf{=}& \frac{\mathbf{3}}{\mathbf{8}}\\ & & \end{array}$

$\begin{array}{rcl}\mathbf{\text{P}}\left(\text{2}\right)\mathbf{\text{:P}}\left(\text{Two} \text{heads}\right)& \mathbf{=}& \frac{\mathbf{\text{Favorable}}\mathbf{ }\mathbf{\text{events}}\mathbf{ }\mathbf{\text{showing}}\mathbf{ }\mathbf{\text{2}}\mathbf{ }\mathbf{\text{heads}}}{\mathbf{\text{Total}}\mathbf{ }\mathbf{\text{events}}\mathbf{ }}\\ & \mathbf{=}& \frac{\mathbf{1}}{\mathbf{8}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{8}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{8}}\\ & \mathbf{=}& \frac{\mathbf{3}}{\mathbf{8}}\\ & & \end{array}$

$\begin{array}{rcl}\mathit{P}\left(3\right)\mathbf{:}\mathit{P}\left( Three heads\right)& \mathbf{=}& \frac{\mathbf{\text{Favorable}}\mathbf{ }\mathbf{\text{events}}\mathbf{ }\mathbf{\text{showing}}\mathbf{ }\mathbf{\text{3}}\mathbf{ }\mathbf{\text{heads}}}{\mathbf{\text{Total}}\mathbf{ }\mathbf{\text{events}}\mathbf{ }}\\ & \mathbf{=}& \frac{\mathbf{1}}{\mathbf{8}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{8}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{8}}\\ & \mathbf{=}& \frac{\mathbf{3}}{\mathbf{8}}\\ & & \end{array}$

c.

The theory of probability distribution is often used by multifarious statisticians in multifarious fields while conducting research.In this case,the researchers try to anticipate all the possible outcomes for a particular set of values.

The associated probabilities of the values of x are plotted on the graph and they are represented by blue bars. In the vertical axis, since the values are in fraction, the first second and third zeroes are values situated between 0 and 1/7.

In this context, probability refers to the chances of the number of heads occuring at a particular event of tossing three coins.This is why the values of x are from 0 to 3, representing the number of heads that can occur at a particular event.

The calculation of the probability of occuring two or three heads is shown below:

$\begin{array}{rcl}\mathit{P}\left(x=2 or x=3\right)\mathbf{ }& \mathbf{=}& \frac{\mathbf{F}\mathbf{a}\mathbf{v}\mathbf{o}\mathbf{r}\mathbf{a}\mathbf{b}\mathbf{l}\mathbf{e}\mathbf{ }\mathbf{ }\mathbf{e}\mathbf{v}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{s}\mathbf{ }\mathbf{ }\mathbf{s}\mathbf{h}\mathbf{o}\mathbf{w}\mathbf{i}\mathbf{n}\mathbf{g}\mathbf{ }\mathbf{ }\mathbf{2}\mathbf{ }\mathbf{ }\mathbf{h}\mathbf{e}\mathbf{a}\mathbf{d}\mathbf{s}}{\mathbf{T}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}\mathbf{ }\mathbf{ }\mathbf{e}\mathbf{ }\mathbf{v}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{s}}\\ & & \mathbf{+}\frac{\mathbf{F}\mathbf{a}\mathbf{v}\mathbf{o}\mathbf{r}\mathbf{a}\mathbf{b}\mathbf{l}\mathbf{e}\mathbf{ }\mathbf{ }\mathbf{e}\mathbf{v}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{s}\mathbf{ }\mathbf{ }\mathbf{s}\mathbf{h}\mathbf{o}\mathbf{w}\mathbf{i}\mathbf{n}\mathbf{g}\mathbf{ }\mathbf{3}\mathbf{ }\mathbf{ }\mathbf{h}\mathbf{e}\mathbf{a}\mathbf{d}\mathbf{s}}{\mathbf{T}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}\mathbf{ }\mathbf{ }\mathbf{e}\mathbf{v}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{s}}\\ & \mathbf{=}& \frac{\mathbf{3}}{\mathbf{8}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{8}}\\ & \mathbf{=}& \frac{\mathbf{4}}{\mathbf{8}}\\ & \mathbf{=}& \frac{\mathbf{1}}{\mathbf{2}}\end{array}$

Thus, the required value is ½.