Question

Toss three fair coins and let x equal the number of heads observed.

1. Identify the sample points associated with this experiment and assign a value of x to each sample point.
2. Calculate p1x2 for each value of x.
3. Construct a graph for p1x2.
4. What is P(x = 2 or x = 3)?

Short answer

a.

Sample Points	Assigned values
HHH	3
HHT	2
HTH	2
THH	2
HTT	1
THT	1
TTH	1
TTT	0

b. 3.36

x	P(x)
1	\(\frac{{\rm{1}}}{{\rm{8}}}\)
2	\(\frac{{\rm{3}}}{{\rm{8}}}\)
3	\(\frac{{\rm{3}}}{{\rm{8}}}\)
4	\(\frac{{\rm{1}}}{{\rm{8}}}\)

c

d. 1/2

Step-by-step solution

Definition of a sample point

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.

Elucidation of the assigned values

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.

Definition of fair coin

b.

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.

Calculation of the probabilities

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

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

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

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

Definition of a probability distribution

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.

Elucidation of the graph

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.

Definition of probability

d.

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.

Calculation

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.