Question

A die is tossed. Let x be the number of spots observed on the upturned face of the die.

a. Find the probability distribution of x and display it in tabular form.

b. Display the probability distribution of x in graphical form.

Short answer

a.

x	1	2	3	4	5	6
P(x)	\(\frac{{\rm{1}}}{{\rm{6}}}\)	\(\frac{{\rm{1}}}{{\rm{6}}}\)	\(\frac{{\rm{1}}}{{\rm{6}}}\)	\(\frac{{\rm{1}}}{{\rm{6}}}\)	\(\frac{{\rm{1}}}{{\rm{6}}}\)	\(\frac{{\rm{1}}}{{\rm{6}}}\)

b.

Step-by-step solution

Definition of a fair die

A fair die has six sides containing numbers from 1 to 6 and is often used in different types of games.The probability of the occurrence of a number is equal since all the numbers have an equal chance of appearing once the die is tossed.

Definition of a probability distribution

The calculation of the probability of each number is shown below:

$\begin{array}{rcl}\mathbf{\text{Probability}}\mathbf{ }\mathbf{\text{of}}\mathbf{ }\mathbf{\text{one number}}& \mathbf{=}& \frac{\mathbf{A}\mathbf{n}\mathbf{y}\mathbf{ }\mathbf{ }\mathbf{o}\mathbf{n}\mathbf{e}\mathbf{ }\mathbf{ }\mathbf{n}\mathbf{u}\mathbf{m}\mathbf{b}\mathbf{e}\mathbf{r}}{\mathbf{T}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}\mathbf{ }\mathbf{ }\mathbf{n}\mathbf{u}\mathbf{m}\mathbf{b}\mathbf{e}\mathbf{r}\mathbf{ }\mathbf{ }\mathbf{o}\mathbf{f}\mathbf{ }\mathbf{n}\mathbf{u}\mathbf{m}\mathbf{b}\mathbf{e}\mathbf{r}\mathbf{s}}\\ \mathbf{\text{Probability}}\mathbf{ }\mathit{o}\mathit{f}\mathbf{ }\mathit{o}\mathit{n}\mathit{e}\mathit{n}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{r}& \mathbf{=}& \frac{\mathbf{1}}{\mathbf{6}}\\ & & \end{array}$

The probability of all the numbers will be$\frac{\mathbf{1}}{\mathbf{6}}$. Based on this data the relevant table is shown below:

x	1	2	3	4	5	6
P(x)	$\frac{\mathbf{1}}{\mathbf{6}}$	$\frac{\mathbf{1}}{\mathbf{6}}$	$\frac{\mathbf{1}}{\mathbf{6}}$	$\frac{\mathbf{1}}{\mathbf{6}}$	$\frac{\mathbf{1}}{\mathbf{6}}$	$\frac{\mathbf{1}}{\mathbf{6}}$

Definition of probability distribution

The probability distribution is a theory often used by researchers while doing statistical analysis.For a particular data generating process, the researchers try to find out anticipate the outcomes for a particular set of values.

Graphical representation

Since the probability of all the values is equal to one-sixth, the size of the bars will be equal as shown in the probability distribution graph below: