Manipur Board Solutions for Chapter: Probability, Exercise 1: EXERCISE 15.1

For a certain locality, the data of monthly household consumption of electricity measured in units of energy is given below:

If one is chosen at random, find the probability that he belongs to the group using the highest amount of energy. Also, find the most probable class of monthly rate of energy consumption to which he/she can belong.

The performance given in grades of $45$ students of a class are as follows:
$({\mathrm{A}}^{+}, \mathrm{A}, {\mathrm{B}}^{+}, \mathrm{B}, {\mathrm{C}}^{+}, \mathrm{C} \mathrm{and} \mathrm{D}$ are $7$ grades$)$

$$\begin{gathered} \mathrm{A}, \mathrm{B}, {\mathrm{C}}^{+}, \mathrm{C}, \mathrm{C}, \mathrm{D}, {\mathrm{A}}^{+}, \mathrm{B}, {\mathrm{B}}^{+}, \mathrm{B}, {\mathrm{B}}^{+}, \mathrm{C}, {\mathrm{C}}^{+}, \mathrm{D}, {\mathrm{A}}^{+} \\ {\mathrm{B}}^{+}, \mathrm{A}, {\mathrm{A}}^{+}, \mathrm{C}, {\mathrm{C}}^{+}, \mathrm{A}, \mathrm{B}, {\mathrm{B}}^{+}, \mathrm{C}, \mathrm{D}, {\mathrm{A}}^{+}, \mathrm{B}, {\mathrm{B}}^{+}, {\mathrm{B}}^{+}, \mathrm{C} \\ \mathrm{B}, {\mathrm{B}}^{+}, \mathrm{B}, {\mathrm{B}}^{+}, {\mathrm{A}}^{+}, \mathrm{C}, \mathrm{C}, {\mathrm{C}}^{+}, \mathrm{A}, {\mathrm{A}}^{+}, \mathrm{B}, {\mathrm{B}}^{+}, {\mathrm{B}}^{+}, \mathrm{C}, \mathrm{A} \end{gathered}$$

Consider two candidates X and Y, such that X says he will get ${\mathrm{A}}^{+}$ and Y says he will get ${\mathrm{C}}^{+}$. Find who is more realistic.

The following is the height and corresponding weight of the members of a sample of $20$ adults drawn at random from a group of parents of the pupils of a school.

A parameter $\zeta$ called body-mass ratio of a person given by $\mathrm{\zeta }=\frac{\mathrm{Weight} \mathrm{in} \mathrm{kg}}{{\left(\mathrm{Height} \mathrm{in} \mathrm{m}\right)}^2}$.

For a person, if $\zeta >25$, he/she considered over weight. Find the probability that a person chosen at random from the sample is not overweight.

A printer declares that his machine has a chance of less than $0.001\%$ of committing typographical mistake of printing misspelt words. In actual counting there were three misspelt words in a dictionary of one million words printed by his machine. Determine whether the printer keeps it or not?

Write answer as yes or no.

In a grouped frequency table of $n$ classes if $p_i$ is the probability of that an observation may belong to $i^{th}$ class with frequency $f_i$. Show that $\sum _{i=1}^n{p}_i=1$. Also for a variable $x$, assuming the constant value $C$ show that $E\left(x\right)=C$.