4 Marks Questions

Question 14 Marks

Define an elementary event.

Answer

What are the meanings of elementary event? The word elementary means simple, non decomposable into elements or other primary constituents and the word event means something that result.
Definition: An elementary event is any single outcome of a trial.
Elementary events are also called simple events.
To illustrate the definition, let us take examples:
$1.$ In the experiment of tossing a coin, the possible outcomes $H$ and $T$.
Any one outcome like $H$ is called an elementary event.
$2.$ In the experiment of rolling a dice, the possible outcomes are $1, 2, 3, 4, 5$ and $6$.
Any one outcome like $4$ is called an elementary event.
Note that $H$ stands for getting a head and $T$ stands for getting a tail in the experiment of tossing a coin.

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Question 24 Marks

Define a trial.

Answer

What is the meaning of trial?
The word trial means a test of performance, qualities, or suitability.
Definition:
Any particular performance of a random experiment is called a trial.
That is, when we perform an experiment it is called a trial of the experiment.
By experiment or trial, we mean a random experiment unless otherwise specified.
Where you are required to differentiate between a trial and an experiment, consider the experiment to be a larger entity formed by the combination of a number of trials.
To illustrate the definition, let us take examples:
$1.$ In the experiment of tossing $4$ coins, we may consider tossing each coin as a trial and therefore say that there are $4$ trials in the experiment.
$2.$ In the experiment of rolling a dice $5$ times, we may consider each rolls as a trial and therefore say that there are $5$ trials in the experiment.
Note that rolling a dice $5$ times is same as rolling $5$ dices each one time. Similarly, tossing $4$ coins is same as tossing one coin $4$ times.

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Question 34 Marks

In Q.No. $7$, what is the probability of getting at least two heads?

Answer

The total number of trials is $200$.
Let $A$ be the event of getting atleast two heads.
The number of times $A$ happens is $72 + 23 = 95$.
Remember the empirical or experimental or observed frequency approach to probability.
If $n$ be the total number of trials of an experiment and $A$ is an event associated to it such that $A$ happens in m-trials.
Then the empirical probability of happening of event $A$ is denoted by $P(A)$ and is given by
$\text{P(A)}=\frac{\text{m}}{\text{n}}$
Therefore, we have $\text{P(A)}=\frac{95}{200}$
$=\frac{19}{40}$

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Question 44 Marks

A die is thrown $100$ times. If the probability of getting an even number is $\frac{2}{5}.$ How many times an odd number is obtained?

Answer

The total number of trials is $100$. Let the number of times an even number is obtained is $x$.
Let $A$ be the event of getting an even number.
The number of times $A$ happens is $x$.
Remember the empirical or experimental or observed frequency approach to probability.
If $n$ be the total number of trials of an experiment and $A$ is an event associated to it such that $A$ happens in m-trials.
Then the empirical probability of happening of event $A$ is denoted by $P(A)$ and is given by
$\text{P(A)}=\frac{\text{m}}{\text{n}}$
Therefore, we have $\text{P(A)}=\frac{\text{x}}{100}.$
But, it is given that $\text{P(A)}=\frac{2}{5}.$
So, we have
$\frac{\text{x}}{100}=\frac{2}{5}$
$\Rightarrow\ 5\text{x}=200$
$\Rightarrow\ \text{x}=\frac{200}{5}$
$\Rightarrow\ \text{x}=40$
Hence an even number is obtained $40$ times.
Consequently, an odd number is obtained $100 - 40 = 60$ times.

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Question 54 Marks

Define an event.

Answer

What are the meanings of event? The word event means something that result.
Definition: An event is a collection of outcomes of a trial of a random experiment.
To illustrate the definition, let us take examples:
$1.$ When two coins are tossed simultaneously, the possible outcomes are $HH, HT, TH$ and $TT$.
Any one outcome like $HH$ is called an event $($elementary event$)$.
The collections like ${HH, HT}, {HH, HT, TT}$ etc are all events $($compound event$)$.
$2.$ In the experiment of rolling a dice, the possible outcomes are $1, 2, 3, 4, 5$ and $6$.
Any one outcome like $4$ is called an event $($elementary event$)$.
The collections like ${1, 2}, {1, 2, 3}, {2, 5, 6}, {2, 3, 4, 5}$ etc are all events $($compound events$).$
Note that $H$ stands for getting a head and $T$ stands for getting a tail in the experiment of tossing a coin.

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Question 64 Marks

The Blood group table of $30$ students of class $IX$ is recorded as follows: $\text{A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O}$ A student is selected at random from the class from blood donation. Find the probability that the blood group of the student chosen is: $1= A\ 2=B\ 3=AB\ 4=O$

Answer

Blood Group	$A$	$B$	$O$	$AB$	Total
Number of Students	$9$	$6$	$12$	$3$	$30$

$1.$ Probability of a student having blood group $A$
$=\frac{\text{Favorable out come}}{\text{Total out come}}$
$=\frac{9}{30}$
$=0.3$
$2.$ Probability of a student having blood group $B$
$=\frac{\text{Favorable out come}}{\text{Total out come}}$
$=\frac{6}{30}$
$=0.2$
$3.$ Probability of a student having blood group $AB$
$=\frac{\text{Favorable out come}}{\text{Total out come}}$
$=\frac{3}{30}$
$=\frac{1}{10}=0.1$
$4.$ Probability of a student having blood group $O$
$=\frac{\text{Favorable out come}}{\text{Total out come}}$
$=\frac{12}{30}=0.4$

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Question 74 Marks

A company selected $2400$ families at random and surveys them to determine a relationship between income level and the number of vehicles in a home. The information gathered is listed in the table below:

Monthly Income $($in $₹)$	Vechicles Per Family
Monthly Income $($in $₹)$	$0$	$1$	$2$	Above $2$
Less than $7000$	$10$	$160$	$25$	$0$
$7000 – 10000$	$0$	$305$	$27$	$2$
$10000 – 13000$	$1$	$535$	$29$	$25$
$13000 – 16000$	$2$	$469$	$29$	$25$
$16000$ or more	$1$	$579$	$82$	$88$

If a family is chosen, find the probability that the family is:
$1.$ Earning $₹\ 10000 - 13000$ per month and owning exactly $2$ vehicles.
$2$. Earning $₹\ 16000$ or more per month and owning exactly $1$ vehicle.
$3.$ Earning less than $₹ 7000$ per month and does not own any vehicle.
$4.$ Earning $₹ 13000 - 16000$ per month and owning more than $2$ vehicles.
$5.$ Owning not more than $1$ vehicle.
$6.$ Owning at least one vehicle.

Answer

$1.$ The probability that the family is earning $10000 - 13000$ and is having exactly $2$ vehicles
$=\frac{\text{No. of families having 10000 - 13000 income and 2 vehicles}}{\text{Total no. of families}}$
$=\frac{29}{2400}$
$2.$ The probability that the family is earning $16000$ or more and is having exactly $1$ vehicle
$=\frac{\text{No. of families having 16000 or more income and 1 vehicle}}{\text{Total no. of families}}$
$=\frac{579}{2400}$
$3.$ The probability that the family is earning less than $7000$ and is having no vehicle
$=\frac{\text{No. of families having less than 7000 income and no vehicle}}{\text{Total no. of families}}$
$=\frac{10}{2400}$
$4.$ The probability that the family is earning $13000-16000$ and is having more than $2$ vehicles
$=\frac{\text{No. of families having 13000 - 16000 income and more than 2 vehicles}}{\text{Total no. of families}}$
$=\frac{25}{2400}=\frac{1}{96}$
$5.$ The probability that the family is having not more than one vehicle
$=\frac{\text{No. of families having not more than 1 vehicle}}{\text{Total no. of families}}$
$=\frac{10+0+1+2+1+160+305+535+469+579}{1200}$
$=\frac{2062}{2400}$
$=\frac{1031}{1200}$
$6.$ The probability that the family is having at least one vehicle
$=\frac{\text{No. of families having at least 1 vehicle}}{\text{Total no. of families}}$
$=\frac{160+305+535+469+579+25+27+29+29+82+0+2+1+25+88}{1200}$
$=\frac{2356}{2400}$
$=\frac{589}{600}$

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Question 84 Marks

Three coins are tossed simultaneously $200$ times with the following frequencies of different outcomes:

Outcome	$3$ heads	$2$ heads	$1$ head	No head
Frequency	$23$	$72$	$77$	$28$

Find the probability of getting at most two heads.

Answer

The total number of trials is $200$.
Let $A$ be the event of getting at most two heads.
The number of times $A$ happens is $28 + 77 + 72 = 177$.
Remember the empirical or experimental or observed frequency approach to probability.
If $n$ be the total number of trials of an experiment and $A$ is an event associated to it such that $A$ happens in m-trials.
Then the empirical probability of happening of event $A$ is denoted by $P(A)$ and is given by $\text{P(A)}=\frac{\text{m}}{\text{n}}$
Therefore, we have $\text{P(A)}=\frac{177}{200}.$

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Question 94 Marks

In a cricket match, a batsman hits a boundary $6$ times out of $30$ balls he plays. Find the probability that a ball played:
$1.$ He hit boundary
$2.$ He does not hit a boundary.

Answer

Number of times the batsman hits a boundary $= 6$
Total number of balls played $= 30$
Number of times the batsman did not hit a boundary $= 30 - 6 = 24$
$1.$ Probability that the batsman hits a boundary $=\frac{\text{Number of times he hit a boundary}}{\text{Total no. of balls}}$
$=\frac{6}{30}$
$=0.2$
$2.$ Probability that the batsman does not hit a boundary $=\frac{\text{Number of times he did not hit a boundary}}{\text{Total no. of balls}}$
$=\frac{24}{30}$
$=0.8$

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Question 104 Marks

A bag contains $4$ white balls and some red balls. If the probability of drawing a white ball from the bag is $\frac{2}{5},$ find the number of red balls in the bag.

Answer

The number of white balls is $4$.
Let the number of red balls is $x$.
Then the total number of trials is $4 + x$.
Let $A$ be the event of drawing a white ball.
The number of times $A$ happens is $4$.
Remember the empirical or experimental or observed frequency approach to probability.
If $n$ be the total number of trials of an experiment and $A$ is an event associated to it such that $A$ happens in m-trials.
Then the empirical probability of happening of event $A$ is denoted by $P(A)$ and is given by
$\text{P(A)}=\frac{\text{m}}{\text{n}}$
Therefore, we have $\text{P(A)}=\frac{4}{4+\text{x}}.$
But, it is given that $\text{P(A)}=\frac{2}{5}.$
So, we have
$\frac{4}{4+\text{x}}=\frac{2}{5}$
$\Rightarrow 2(4 + x) = 20$
$\Rightarrow 8 + 2x = 20$
$\Rightarrow 2x = 20 - 8$
$\Rightarrow 2x = 12$
$\Rightarrow\ \text{x}=\frac{12}{2}$
$\Rightarrow x = 6$
Hence the number of red balls is $6$.

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Question 114 Marks

Given below is the frequency distribution of wages $($in $₹)$ of $30$ workers in certain factory:

Wages$($in $₹)$	$110-130$	$130-150$	$150-170$	$170-190$	$190-210$	$210 -230$	$230-250$
No of workers	$3$	$4$	$5$	$6$	$5$	$4$	$3$

A worker is selected at random. Find the probability that his wages are:
$1.$ Less than $₹ 150$
$2.$ At least $₹ 210$
$3.$ More than or equal to $150$ but less than $₹ 210$

Answer

Total number of workers $= 30$
$1.$ Probability that the worker wages are less than $₹ 150$
$=\frac{\text{No. of workers having wages below ₹ 150}}{\text{Total no. of workers}}$
$=\frac{3+4}{30}=\frac{7}{30}$
$2.$ Probability that the worker wages are at leas $₹.210$
$=\frac{\text{No. of workers having wages below ₹ 210}}{\text{Total no. of workers}}$
$=\frac{4+3}{30}=\frac{7}{30}$
$3.$ Probability that the worker wages are more than or equal to $₹ 150$ but less than $₹ 210$
$=\frac{\text{No. of workers having wages more than $₹ 150$ but less than ₹ 210}}{\text{Total no. of workers}}$
$=\frac{5+6+5}{30}=\frac{16}{30}=\frac{8}{15}$

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