3 Marks Question

Question 13 Marks

Two coins are tossed simultaneously $500$ times with the following frequencies of different outcomes: Two heads: $95$ times One heads: $290$ times No heads: $115$ times Find the probability of occurrence of each of these events.

Answer

Probability $(E)$ $=\frac{\text{Number of trialsin which events happen}}{\text{Total no. of trials}}$
$P$(getting two heads) $=\frac{95}{500}=0.19$
$P$(getting one tail) $=\frac{290}{500}=0.58$
$P$(getting no head) $=\frac{115}{500}=0.23$

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Question 23 Marks

To know the opinion of the students about Mathematics, a survey of $200$ students was conducted. The data is recorded in the following table:

Opinion	Like	Dislike
Number of students	$135$	$65$

Find the probability that a student chosen at random:
$1.$ Likes Mathematics
$2.$ Does not like it.

Answer

$1.$ Probability that a student likes mathematics
$=\frac{\text{Favorable out come}}{\text{Total out come}}$
$=\frac{135}{200}$
$=0.675$
$2.$ Probability that a student does not like mathematics
$=\frac{\text{Favorable out come}}{\text{Total out come}}$
$=\frac{65}{200}$
$=0.325$

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Question 33 Marks

The percentage of marks obtained by a student in the monthly unit tests are given below:

Unit Test	$I$	$II$	$III$	$IV$	$V$
Percentage of Mark Obtained	$69$	$71$	$73$	$68$	$76$

Find the probability that the student gets:
$1.$ More than $70 \%$ marks.
$2.$ Less than $70 \%$ marks.
$3.$ A distinction

Answer

$1.$ Let $E$ be the event of getting more than $70 \%$ marks.
No of times $E$ happens $= 3$
Probability$($getting more than $70 \%)$
$=\frac{\text{Number of times student got more than 70}}{\text{Total no. of exams taken}}$
$=\frac{3}{5}=0.6$
$2.$ Let $F$ be the event of getting less than $70 \%$ marks
No of times $F$ happen $= 2$
Probability$($getting more than $70 \%)$
$=\frac{\text{Number of times student got more than 70}}{\text{Total no. of exams taken}}$
$=\frac{2}{5}=0.4$
$3.$ Let $G$ be the event of getting distinction
No of times $G$ happen $= 1$
Probability$($getting distinction$)$
$=\frac{\text{Number of times student got distinction}}{\text{Total no. of exams taken}}$
$=\frac{1}{5}=0.2$

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Question 43 Marks

Eleven bags of wheat flour, each marked $5\ kg$, actually contained the following weights of flour (in Kg).

$4.97$

$5.05$

$5.08$

$5.03$

$5$

$5.06$

$5.08$

$4.98$

$5.04$

$5.07$

$5$

Find the probability that any of these bags chosen at random contains more than $5\ kg$ of flour.

Answer

Number of bags weighting more than $5\ kg = 7$
Total no of bags $= 11$
Probability of having more than $10\ kg$ of rice
$=\frac{\text{No. of bages weighting more than 5kg}}{\text{Total no. of bages}}$
$=\frac{7}{11}=0.63$

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Question 53 Marks

A coin is tossed $1000$ times with the following frequencies: Head: $455$, Tail = $545.$ Compute the probability of each event.

Answer

It is given that the coin is tossed $1000$ times.
The number of trials is $1000$.
Let us denote the event of getting head and of getting tails be $E$ and $F$ respectively.
Then, Number of trials in which the $E$ happens $= 455$
So, Probability of $E$
$=\frac{\text{Number of even theads}}{\text{Total no. of trials}}$
$\text{i.e}.\text{P(E)}=\frac{455}{1000}=0.455$
Similarity, the probability of the event getting a tail
$=\frac{\text{Number of tails}}{\text{Total no. of trials}}$
$\text{i.e.}\text{P(F)}=\frac{545}{1000}=0.545$

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