1 Marks Each

Question 11 Mark

Two balanced coins are tossed simultaneously. Write the sample space of this random experiment.

Answer

We shall consider any one of the two coins here as the ﬁrst coin and the other as the second coin. If we denote the head as $H$ and the tail as $T,$ the sample space will be as follows : $U = \{HH, HT, TH, TT\}$

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Question 21 Mark

State the limitations of statistical definition of probability.

Answer

The limitations of statistical definition of probability are as follows :

The infinite value of n cannot be taken in practice.
The exact value of probability cannot be known.

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Question 31 Mark

State the assumptions of mathematical definition of probability.

Answer

The assumptions of mathematical definition of probability are as follows :

The number of outcomes in the sample space is finite.
The number of all possible outcomes of the sample space is known.
The outcomes of the sample space are equi-probable.

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Question 41 Mark

State the characteristics of random experiment.

Answer

The characteristics of random experiment are as follows :

It can be repeated under identical conditions,
Its all possible outcomes are known,
It cannot be predicted with certainty which outcome will appear and
It results into a certain outcome.

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Question 51 Mark

Give the illustrations of impossible and certain event.

Answer

Impossible event: Event to get the number greater than 6 on the upper side of a balanced die.
Certain event : Event to get head or tail in tossing a balanced coin.

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Question 61 Mark

Interpret $P(A|B)$ and $P(B|A).$

Answer

Interpretation of $P (A|B)$ The conditional probability of the event $A$ under the condition that the event $B$ is occurred. Interpretation of $P (B|A):$ The conditional probability of the event $B$ under the condition that the event $A$ has occurred.

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Question 71 Mark

Write the law of multiplication of probability for two independent events $A$ and $B$ in sample space.

Answer

If $A$ and $B$ are the any two events of a finite sample space $U$. i.e. $P(A/B) = P(A)$ and $P(B/A) = P(B),$ for two independent events law of multiplication of probability is written as under.
$P(A/B) = P(A) ∙ P(B)$

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Question 81 Mark

Define independent events.

Answer

A and B are any two events of a finite sample space

.
The event B is called independent of A if the probability of occurrence of event A does not effect the probability of occurrence of the event B i.e. if P(A/B) = P(A) and P(B/A) = P(B), then A and B are called the mutually independent event.

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Question 91 Mark

State the formula for the probability of occurrence of at least one event out of three events $A, B$ and $C.$

Answer

The formula for the probability of occurrence of at least one event out of three events $A, B$ and $C$ is as follows :
$P(A∪B∪C) = P(A) + P(B) + P(C)-P(A∩B) -P(A∩C)-P(B∩C)+P(A∩B∩C)$

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Question 101 Mark

Define conditional probability.

Answer

$U$ is a finite sample space and $A$ and $B$ are any two events of $U.$ The probability of event $B,$ under the condition that event $A$ is happened, is called conditional probability of the event $B.$

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Question 111 Mark

Write the sample space of a random experiment of throwing a balanced die and a balanced coin simultaneously.

Answer

The sample space of a random experiment of throwing a balanced die and a balanced coin simultaneously are as follows :
$U = \{(1,H), (2,H), (3,H), (4,H), (5,H), (6,H), (1,T), (2,T), (3,T), (4,T), (5,T), (6,T)\}$

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Question 121 Mark

Define an event.

Answer

Any subset of sample space of a random experiment is called an event.
If is denoted by the letters $A, B, C, \ldots$. or $A_1, A_2, A_3, \ldots$.

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Question 131 Mark

$1998$ tickets out of $2000$ tickets do not have a prize. If a person randomly selects one ticket from $2000$ tickets, then what is the probability that the ticket selected is eligible for prize$ ?$

Answer

Total number of tickets $=2000$
No. of tickets do not have a prize $=1998$
No. of tickets eligible for prize $=2000-1998$
$
=2
$
Total number of outcomes of selecting a ticket is $n={ }^{2000} c_1=2000$
$A=$ Event that the selected ticket is eligible for prize
$
\begin{aligned}
& \therefore \mathrm{m}={ }^2 c_1=2 \\
& \therefore \mathrm{P}(\mathrm{A})=\frac{m}{n} \\
& =\frac{2}{2000} \\
& =\frac{1}{1000}
\end{aligned}
$

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Question 141 Mark

For two independent everis $A$ and $B, P(B \mid A)=\frac{1}{2}$ and $P(A \cap B)=\frac{1}{5}$, Find $P(A)$

Answer

$P ( B \mid A )=\frac{1}{2}, P ( A \cap B )=\frac{1}{5}, P ( A )=?$
$P ( B \mid A )=\frac{P(A \cap B)}{P(A)}$
$\therefore \frac{1}{2}=\frac{\frac{1}{5}}{ P ( A )}$
$\therefore \frac{1}{2} P ( A )=\frac{1}{5}$
$\therefore P ( A )=\frac{1}{5} \times 2=\frac{2}{5}$

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Question 151 Mark

State the number of sample points in the random experiment of tossing one balanced coin and two balanced dice simultaneously.

Answer

The number of sample points in the random experiment of tossing one balanced coin and two balanced dice simultaneously is $\mathrm{n}=2^{\prime} \quad 6^2=2 \quad 36=72$.

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Question 161 Mark

State the number of sample points in the random experiment of tossing five balanced coins.

Answer

The number of sample points in the random experiment of tossing five balanced coins is $n=2^5=32$.

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Question 171 Mark

$2\%$ items in a lot are defective. What is the probability that an item randomly selected from this lot is non defective?

Answer

$A=$ Event that an item is defective
$
\therefore \mathrm{P}(\mathrm{A})=2 \%=\frac{2}{100}=0.02
$
$\mathrm{A}^{\prime}$ = Event that an item is non-defective
$
\therefore P\left(A^{\prime}\right)=1-P(A)=1-0.02=0.98
$

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Question 181 Mark

Two events $A$ and $B$ in a sample space are mutually exclusive and exhaustive. If $P(A)=\frac{1}{3}$, find $P(B)$.

Answer

$A$ and $B$ are mutually exclusive and exhaustive events.
$
\begin{aligned}
& \therefore P(A)+P(B)=1\left(\text { Putting } P(A)=\frac{1}{3}\right) \\
& \frac{1}{3}+P(B)=1 \\
& \therefore P(B)=1-\frac{1}{3}=\frac{2}{3}
\end{aligned}
$

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Question 191 Mark

$P (A’ ∩ B) = 0.45$ and $A ∩ B = ∅$, find $P(B).$

Answer

$P (A’ ∩ B) = 0.45$ and $A ∩ B = \Phi $ are given.
$\therefore P(A ∩ B) = 0$
Now, $P(A’ ∩ B) = P(B) – P(A ∩ B)$
$\therefore 0.45 = P(B) – 0$
$\therefore P(B) = 0.45$

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Question 201 Mark

Draw the Venn diagram for $A-B,$ the difference event of $A$ and $B.$

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Question 211 Mark

If $P(A)=0.3$ and $P(A \cap B)=0.03,$ Find $P(B/A).$

Answer

$P(A)=0.3, P(A \cap B)=0.03$ are given.
$
\therefore \mathrm{P}(\mathrm{B} \mid \mathrm{A})=\frac{P(A \cap B)}{P(A)}=\frac{0.03}{0.3}=0.1
$

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Question 221 Mark

If $A=\{x \mid 0<x<1\}$ and $B=\left\{x \mid \frac{1}{4} \leq x \leq 3\right\}$, then find $A \cap B$.

Answer

$\begin{aligned} & A=\{x \mid 0<x<1\}=\left\{\frac{1}{4}, \frac{1}{2}, \frac{3}{4}\right\} \\ & B=\left\{x \mid \frac{1}{4} \leq x \leq 3\right\}=\left\{\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, \ldots \ldots, 1,2,3\right\} \\ & \therefore A \cap B=\left\{x \mid \frac{1}{4} \leq x<1\right\}\end{aligned}$

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Question 231 Mark

Define: Certain Event.

Answer

The special subset $U$ of the sample space of random experiment is called a certain event.
The certain event is an event which always occurs.
It is denoted by $U.$

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Question 241 Mark

Define: Impossible Event:

Answer

The special subset $∅$ or $\{\ \}$ of the sample space of a random experiment is called an impossible event.
Impossible event is an event which never occurs.
It is denoted by $∅$ or $\{\ \}.$

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Question 251 Mark

Define: Probability $($Statistical definition$):$

Answer

Suppose, a random experiment is repeated $\mathrm{n}$ times under identical conditions. If an 'event A occurs in $\mathrm{m}$ trials then the relative frequency $\frac{m}{n}$ of the event A gives the estimate of the probability of the event $A$. When $n$ tends to infinity, the limiting value of $\frac{m}{n}$ is called the probability of the event A. Thus,
$
P(A)=\lim _{n \rightarrow \infty} \frac{m}{n}
$

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Question 261 Mark

Define: Favorable Outcomes.

Answer

If some primary outcomes out of all the primary outcomes of a random experiments indicates the occurrence of an event $A,$ then these outcomes are said to be favorable to then occurrence of the event $A.$

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Question 271 Mark

Define: Equi-probable Events.

Answer

If there is no apparent reason to believe that out of one or more events of a random experiment, any one event is more or less likely to occure than other events, then those events are called equiprobable events.

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Question 281 Mark

Define: Sample Space.

Answer

The set of all possible outcomes of a random experiment is called a sample space of that experiment. It is generally denoted by the symbol or $S.$

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Question 291 Mark

Define: Random Experiment

Answer

The experiment which can be independently repeated under indentical conditions and all its possible outcomes are known but which of the outcomes will appear cannot be predicted with certainly before conducting the experiment is called a random experiment.

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Question 301 Mark

Give two example of random experiment.

Answer

Two examples of random experiment are :

The experiment of throwing a balanced die and
The experiment of finding defective units from a lot of units produced.

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Question 311 Mark

A balanced coin is tossed three times. Find the probability of the following events: Getting more number of heads than tails

Answer

$\frac{1}{2}$

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Question 321 Mark

A balanced coin is tossed three times. Find the probability of the following events: Getting head and tail alternately

Answer

$\frac{1}{4}$

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Question 331 Mark

A balanced coin is tossed three times. Find the probability of the following events:Getting less than two heads

Answer

$\frac{1}{2}$

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Question 341 Mark

A balanced coin is tossed three times. Find the probability of the following events:Getting at the most one head

Answer

$\frac{1}{2}$

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Question 351 Mark

A balanced coin is tossed three times. Find the probability of the following events:Getting more than one head

Answer

$\frac{1}{2}$

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Question 361 Mark

A balanced coin is tossed three times. Find the probability of the following events:Getting at least one head

Answer

$\frac{7}{8}$

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Question 371 Mark

A balanced coin is tossed three times. Find the probability of the following events: Not getting a single head

Answer

$\frac{1}{8}$

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Question 381 Mark

A balanced coin is tossed three times. Find the probability of the following events: Getting all three heads

Answer

$\frac{1}{8}$

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Question 391 Mark

Write the sample space for the experiment of randomly selecting three numbers from the first five natural numbers.

Answer

$U = \{(1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 3, 4), (1, 3, 5), (1, 4, 5), (2, 3, 4), (2, 3, 5), (2, 4, 5), (3, 4, 5)\}.$

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Question 401 Mark

A balanced coin in thrown in a random experiment till the first head is obtained. The experiment is terminated with a trial of first head. Write the sample space of this experiment and state whether it is finite or Infinite.

Answer

$U=\{H, TH, TTH, TTTH, ...\}$ The sample space of this experiment is infinite.

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Question 411 Mark

Write the sample space for randomly selecting one minister and one deputy minister from four persons.

Answer

$U =\{(a, b), (a, c), (a, d), (b, a), (b, c), (b, d), (c, a), (c, b), (c, d), (d, a), (d, b), (d, c)\}.$

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Question 421 Mark

Write the sample space for the marks $($in Integers$)$ scored by a student appearing for an examination of $100$ marks and state the number of sample points in it.

Answer

$U=\{0, 1, 2, 3 …,100\}$ The number of sample points in the sample space $U=101.$

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Question 431 Mark

State the sample space for the following random experiments: Two persons are to be selected from five persons $a, b, c, d, e.$

Answer

$U = \{(a, b), (a, c), (a, d), (a, e), (b,c), (b, d), (b, e), (C, d), (c, e), (d, e)\}$

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Question 441 Mark

State the sample space for the following random experiments: A balanced die with six sides and a balanced coin are tossed together.

Answer

$U = \{(1, H), (2, H), (3, H), (4, H), (5, H (6, H), (1, T), (2, T), (3, T), (4, T), (5, T (6,T)\}$

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Question 451 Mark

State the sample space for the following random experiments: A balanced coin is thrown three times.

Answer

$U =\{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$

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Question 461 Mark

Write a sample space for a coin tossed four times.

Answer

Sample space for a coin tossed four times,
$U =\{HHHH; HHHT; HHTT; HTTT; HHTH; HTHH; HTHT; HTTH; TTTT; TTTH;$$TTHH; THHH; TTHT; THTT; THTH; THHT\}$

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Question 471 Mark

State the formula for the probability of occurrence of at least one event out of three events $A, B,$ and $C.$

Answer

Occurrence at least one event out of three event $A, B$ and $C$ or $A \cup B \cup C.$
$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$

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Question 481 Mark

Arrange in descending order: $P(A), P(A \cap B), P(A \cup B)$.

Answer

$P(A) \geq P(A \cap B)$ and $P(A \cup B) \geq P(A)$. Hence. the required descending order is $P(A \cup B), P(A), P(A \cap B)$.

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Question 491 Mark

If $A$ and $B$ are exhaustive events, what is the value of $P(A \cup B) ?$

Answer

If $A$ and $B$ are exhaustive events, $P(A \cup B)=1$.

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Question 501 Mark