Question 511 Mark
For events $A$ and $B$, if $A \cap B-0$, then $A$ and $B$ are events of which type?
AnswerFor events $A$ and $B$, if $A \cap B=0$, then events $A$ and $B$ are called mutually exclusive events.
View full question & answer→Question 521 Mark
Write the sample space of marks obtained in an examination of $100$ marks.
AnswerIn an examination of $100$ marks one can get minimum zero mark and maximum $100$ marks. Hence, the sample space of marks obtained is $U=\{d, 1,2,398, 99, 100\}.$
View full question & answer→Question 531 Mark
Write the sample space if two letters are written simultaneously out of two postcards, two inlands and two airletters.
AnswerLet us denote $P=$ postcard, $\mid=$ inland and $A=$ airetters, If two letters are written simultaneously, its sample space obtained is $U=\{P E \|, A A, P I, P A, I A\}$.
View full question & answer→Question 541 Mark
Which primary outcomes are called favourable outcomes?
AnswerThe primary outcomes, out of all the primary outcomes of a random experiment, that indicate the occurrence of an event $A$ are said to be favourable outcomes for the occurrence of the event $A$
View full question & answer→Question 551 Mark
Define. Elementary events
AnswerEvents consisting of only a single element of a sample space $U$ are called elementary events.
View full question & answer→Question 561 Mark
Write the definition of an infinite sample space.
AnswerA sample space whose elements are not finite is called an infinite sample space.
View full question & answer→Question 571 Mark
What is a finite sample space? Give Its one illustration.
AnswerA sample space having a finite number of elements is called a finite sample space. The sample space obtained by throwing a balanced die $U=\{1,2,3,4,5,6\}$ is an example of a finite sample space.
View full question & answer→Question 581 Mark
"Adding 2 to 2 , we get $4^{\circ}$ can this event be called a random event? Give reason.
Answer"Adding 2 to 2 , we get $4^{*}$ is not a random event, because this event is not based on chance.
View full question & answer→Question 591 Mark
On what principle is the science of statistics based?
AnswerThe science of statistics is based on the theory of probability.
View full question & answer→Question 601 Mark
AnswerAn event depending on chance is called a random event.
View full question & answer→Question 611 Mark
Which word Is used for the possibility of occurrence of a random event?
AnswerThe word probability is used for the possibility of occurrence of a random event.
View full question & answer→Question 621 Mark
Write the sample space of a random experiment of throwing one balanced die and a balanced coin simultaneously.
AnswerThe sample space of a random experiment of throwing one balanced die and a balanced coin simultaneously is obtained as follows: $U = \{(1, H), (2, H), (3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T), (5, T), (6, T)\}$
Where, $H =$ Head; $T =$ Tail; $1, 2, 3, 4, 5, 6=$ Numbers on die.
View full question & answer→Question 631 Mark
Give two examples of random experiment.
AnswerTwo examples of random experiment are: $(1)$ The experiment of throwing a balanced die and $(2)$ The experiment of finding defective units from a lot of units produced.
View full question & answer→Question 641 Mark
Write the sample space for randomly selecting one minister and one deputy minister from four persons.
Answer$U = \{(a, b), (a, C), (a, d), (b, a), (b, C), (b, d), (c, a), (c, b), (c, d), (d, a), (d, b), (d, c)\}$
View full question & answer→Question 651 Mark
State the sample space for the following random experiments: A balanced coin is thrown three times.
Answer$U = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$
View full question & answer→Question 661 Mark
What is the value of $P(B)$ if $P(A/B)=0.36$ and $P(A \cap B)=0.18?$
Answer
| $P(A \cap B) = P(A/B) ∙ P(B)$ |
| $\therefore 0.18 = 0.36\times P(B)$ |
| $\therefore P(B)=\frac{0.18}{0.36}=0.5$ |
| $\therefore$ The value of $P(B) = 0.5.$ |
View full question & answer→Question 671 Mark
If $A$ and $B$ are mutually exclusive and exhaustive events then, $P (A) =0.30$ and $P (B) =0.45.$ Is it possible? Give reason.
Answer
| $A$ and $B$ are mutually exclusive and exhaustive events |
| $\therefore (A \cup B) = U$ and $(A \cap B) = ∅$ |
| $\therefore P(A \cup B) = P(\cup)$ and $P(A \cap B) = 0$ |
| $\therefore P(A) + P(B) = 1$ |
| But, here $P(A) + P(B) = 0.30 + 0.45 = 0.75.$ |
| It is not equal to $1$ therefore it is not possible. |
View full question & answer→Question 681 Mark
If $A$ and $B$ are mutually exclusive events then, $P (A) =0.65$ and $P (A \cap B’) =0.50,$ is this possible? Give reason.
Answer
| $A$ and $B$ are mutually exclusive events |
| Hence, $(A \cap B) = ∅$ |
| $\therefore A = (A \cap B’) \cup (A \cap B)$ |
| $\therefore A = A \cap B’$ |
| $\therefore P(A) = P(A \cap B’)$ |
| But, here $P(A) = 0.65$ and $P(A \cap B') = 0.50.$ |
| Hence, $P(A) ≠ P(A \cap B'), P(A) = 0.65$ and $P(A \cap B') = 0.50.$ |
| So it is not possible. |
View full question & answer→Question 691 Mark
If three events are mutually independent events $A, B$ and $C$ then, write the formula to find probability $(i)$ occurrence of three events and $(ii)$ occurrence of any two events.
Answer$A, B$ and $C$ are mutually independent event.
$(i)$ Occurrence of three events $A \cap B \cap C.$
$∴ P(A \cap B \cap C) = P(A) ∙ P(B) ∙ P(C)$
$(ii)$ Occurrence of any two events $A \cap B \cap C'$ or event $A \cap B' \cap C$ or event $A' \cap B \cap C$
$∴ P[(A \cap B \cap C') \cup (A \cap B' \cap C) \cup (A' \cap B \cap C)] = P(A \cap B \cap C') + P(A\cap B' \cap C) + P(A' \cap B \cap C)$
View full question & answer→Question 701 Mark
For the two mutually exclusive and exhaustive events $A$ and $B,$ if $P(A)=1/3;$ then what is the value of $P(B)?$
Answer
| Events $A$ and $B$ are mutually exclusive and exhaustive events. |
| $\therefore A \cup B = U$ |
| $\therefore P(A \cup B) = U$ |
| $\therefore P(A) + P(B) = 1$ |
| $\therefore \frac{1}{3} + P(B) = 1$ |
| $\therefore P(B) = 1 - \frac{1}{3}$ |
| $\therefore P(B) = \frac{2}{3}$ |
| $\therefore P(B) = \frac{2}{3}$ |
View full question & answer→Question 711 Mark
If $P(A)=0.6; P(B)=0.7$ and $P(A \cup B)=0.9$ then find the probability of difference event $A-B.$
Answer
| $P(A - B)$ |
$= P(A \cup B) - P(B)$ |
| |
$= 0.9 - 0.7$ |
| |
$= 0.2$ |
| $\therefore P(A-B) = 0.2$ |
View full question & answer→Question 721 Mark
For two events $A$ and $b$ of the finite sample space $U,$ if $P(A)=P(A/B),$ then can event $A$ and $B$ called independent event $?$ Give reason for your answer.
Answer
| For two events $A$ and $B$ of the finite sample space $\cup, P(A) = (A/B).$ |
| Hence, events $A$ and $B$ is an independent event. |
| Event $A$ is not dependent on event $B.$ |
| Hence, $P(A \cap B) = P(A/B) ∙ P(B) = P(A) ∙ P(B).$ |
View full question & answer→Question 731 Mark
Prove that $P (A) +P (A’) =1.$
Answer
| $A\cup A' = \cup$ and $A\cap A' = \emptyset$ |
| $P(A\cup A') = P(\cup)$ |
| $P(A) + P(A)’ = 1$ |
View full question & answer→Question 741 Mark
If $(A-B)=0.20; P(B-A)=0.25$ and $P(A \cap B)=0.35$ the find $P(A \cup B).$
Answer
| As we know $($from ven diagram$)$ that, |
| $A \cup B = (A-B) \cup (A \cap B) \cup (B-A)$ |
| $\therefore P(A \cup B) = P(A-B) + P(A \cap B) + P(B-A)$ |
| $= 0.20 + 0.25 + 0.35 = 0.80$ |
| $\therefore P(A \cup B) = 0.80$ |
View full question & answer→Question 751 Mark
If events $A, B$ and $C$ are mutually exclusive and exhaustive events and $P(A) = \frac{1}{5}$ and $P(B) = \frac{2}{3}$ then find the value of $P(C)?$
Answer
| $A, B$ and $C$ are mutually exclusive and exhaustive events |
| $\therefore P(A) + P(B) + P(C) = 1$ |
| $\therefore \frac{1}{5} + \frac{2}{3} + P(C) = 1$ |
| $\therefore P(C) =1-\frac{1}{5}-\frac{2}{3}=\frac{15-3-10}{15}=\frac{2}{15}$ |
| $\therefore $ Value of $P(C) = \frac{2}{15}$ |
View full question & answer→Question 761 Mark
Write complementary event of intersection event of $ A\cap B.$
AnswerComplementary event of intersection event of $A \cap B$ is $A' \cup B'.$
View full question & answer→Question 771 Mark
If $P (A) +P (B) +P(C) = 1,$ then what can you say about events $A, B$ and $C?$
AnswerIf $P(A) + P(B) + P(C) = 1$ then events $A, B$ and $C$ are exhaustive and mutually exclusive events.
View full question & answer→Question 781 Mark
If $A$ and $B$ are exhaustive events, then what is the value of $P (A \cup B)?$
AnswerIf $A$ and $B$ are exhaustive events, then $P(A \cup B) = 1.$
View full question & answer→Question 791 Mark
For events $A$ and $b,$ if $A \cup B = U$ then $A$ and $b$ are events of which type$?$
AnswerFor events $A$ and $B,$ if $A \cup B = U,$ then events $A$ and $B$ are called exhaustive events.
View full question & answer→Question 801 Mark
Interpret event $A \cap B =\emptyset.$
AnswerEvent $A \cap B = \emptyset$ means event $A$ and $B$ do not occur together.
View full question & answer→Question 811 Mark
If $A \cap B=\emptyset$ and $P(A \cap B’)= 0.48,$ then find $P(A).$
Answer
| $P(A)$ |
$= P(A \cap B') + P(A \cap B)$ |
|
| |
$= 0.48 + 0$ |
| |
$= 0.48$ |
| $\therefore$ Value of $P(A) = 0.48.$ |
|
View full question & answer→Question 821 Mark
For the independent event $A$ and $B,$ if $P(A)=0.4$ and $P(B) =0.6$ then what is the value of $P(A \cap B)?$
Answer
| $P(A \cap B)$ |
$= P(A) ∙ P(B)$ |
| |
$= 0.4\times 0.6$ |
| |
$= 0.24$ |
| $\therefore$ Value of $P(A \cap B)$ is $0.24.$ |
View full question & answer→Question 831 Mark
If $P (A) = 0.35$ and $P(A \cap B’) =0.06$ then find the value of $P(A \cap B) .$
Answer
| $P(A \cap B’) = P(A) - P(A \cap B)$ |
| $\therefore 0.06 = 0.35 - P(A \cap B)$ |
| $\therefore P(A \cap B) = 0.35 - 0.06 = 0.29$ |
| $\therefore$ Value of $P(A \cap B) = 0.29$ |
View full question & answer→Question 841 Mark
“Value of probability $P(A)$ is always between $-1$ and $1.$” Correct the statement.
AnswerThe value of probability $P(A)$ is always between $0$ and $1.$
$0\leq P(A) \leq 1.$
View full question & answer→Question 851 Mark
“If $A⊂B$ the $P(A) \geq P(B)$”. Correct the statement.
AnswerIf $A \subset B$ then $P(A) \leq P(B).$
View full question & answer→Question 861 Mark
For event $A$ and $B, A \subset B.$ If $P(A) =0.3$ and $P(B) =0.5$ the find the probability of event $B-A.$
Answer
| If for event $A$ and $B, A \subset B$ then $P(A) \leq P(B)$ and |
| $P(B-A) = P(B) - P(A).$ |
| $\therefore P(B - A) = 0.5 - 0.3 = 0.2$ |
| $\therefore $ The probability of event $B - A$ is $0.2.$ |
View full question & answer→Question 871 Mark
Two events $A$ and $B$ are mutually exclusive and exhaustive. If the probability of event $A$ is twice the probability of $B$ then what is the probability of event $B?$
Answer
| Events $A$ and $B$ are mutually exclusive and exhaustive. Hence, $P(A) + P(B) = 1$ and event $A$ is twice the event $B$ therefore $P(A) = 2P(B).$ |
| $P(A) + P(B) = 1$ |
| $2P(B) + P(B) = 1 [$Here $P(A) = 2P(B)]$ |
| $3P(B) = 1$ |
| $P(B) = 1/3$ |
| The probability of event $B$ is $\frac{1}{3}$. |
View full question & answer→Question 881 Mark
What is the value of $P (B)$ if events $A$ and $B$ are mutually exclusive and exhaustive and $P(A) = 2P(B)?$
Answer
| If events $A$ and $B$ are mutually exclusive and exhaustive than, |
| $P(A) + P(B) = 1$ |
| $\therefore 2P(B) + P(B) = 1 [$here putting $P(A) = 2P(B)]$ |
| $\therefore 3P(B) = 1$ |
| $\therefore P(B) = \frac{1}{3}$ |
| $\therefore$ The value of $P(B)$ is $\frac{1}{3}$ |
View full question & answer→Question 891 Mark
Find $A \cup B$ if $A =\{x/0\leq x<5\}.$
Answer
| $A=\{x/O < x < 3\}$ |
| $B = \left\{x / \frac{1}{2} \leq x<5\right\}$ |
| $A \cup B=\{x / 0 \leq x<5\}$ |
View full question & answer→Question 901 Mark
Find $A \cap B$ if $A =\{x/0 < x < 2\}$ and $B = \{x/1/2 \leq x < 3\}.$
Answer
| $A= \{x/O < x < 2\}$ |
| $B = \left\{x / \frac{1}{2} \leq x<3\right\}$ |
| $\therefore A \cap B = \{x/ \frac{1}{2}\leq x< 2\}$ |
View full question & answer→Question 911 Mark
If the probability of at least one of the two events occurs is $0.85,$ then what is the probability of that no event occur?
Answer
| Here, $P(A \cup B) = 0.85,$ |
| Hence, $P(A' \cap B') = 1 - P(A \cup B) = 1 - 0.85 = 0.15$ |
| $\therefore P(A' \cap B') = 0.15$ |
| $\therefore$ The probability of no event occur is $0.15.$ |
View full question & answer→Question 921 Mark
State the statistical definition of probability.
Answer
| $P(A) = \lim _{n \rightarrow \infty} \frac{m}{n}$ |
| as $n$ tends to infinity, the limiting value of ratio of $\frac{m}{n}$ is taken. |
View full question & answer→Question 931 Mark
For two events $A$ and $B$ of the finite sample space $P(A)=0.80$ and $(A \cup B)=0.35.$ Correct the statement.
Answer
| For two events $A$ and $B$ of the finite sample space $P(A) = 0.80$ and $P(A \cup B) = 0.35.$ This statement is wrong because |
| $P(A) \leq P(A \cup B),$ here $P(A \cup B) < P(A).$ |
View full question & answer→Question 941 Mark
State the D’morgan’s law of probability.
AnswerD’morgan’s law of probability is:
$P(A' \cup B') = P(A \cap B)‘ = 1 - P(A \cap B)$
$P(A' \cap B') = P(A \cup B)'= 1 - P(A \cup B)$
View full question & answer→Question 951 Mark
If $A$ and $B$ are independent event, write the formula of $P (A \cap B).$
AnswerIf $A$ and $B$ are independent event then $P(A \cap B) = P(A) . P(B).$
View full question & answer→Question 961 Mark
For mutually exclusive events $A$ and $B$ if $A \cup B=U$ and $P(A) = \frac{2}{5}$ the find $P(B).$
Answer
| $A \cup B = U$ |
| $\therefore P(A \cup B) = P(\cup)$ |
| $\therefore P(A) +P(B) = 1$ |
| $\therefore \frac{2}{5} + P(B) = 1$ |
| $\therefore P(B) =1-\frac{2}{5}=\frac{3}{5}$ |
| $\therefore$ The value of $P(B)$ is $\frac{3}{5}.$ |
View full question & answer→Question 971 Mark
For two events $A$ and $b$ of the finite sample space, state the law of conditional probability?
Answer
| Following is the law of conditional probability or two events $A$ and $B$ |
$P(B/A) = \frac{P(A \cap B)}{P(A)}$ OR
$P(A/B) = \frac{P(A \cap B)}{P(A)}$ |
| Where, $P(A) \neq 0$ and $ P(B)\neq 0$ |
View full question & answer→Question 981 Mark
If events $A= \{0, 1, 2, 3, 4, 5\}$ and $B=\{x/0 \leq x<3.5\}, x \in{Z}$ then write event $A \cap B.$
Answer$A = \{0, 1, 2, 3, 4, 5\}; B = \{ \chi /0 \leq\chi <3.5\}; \chi \in Z = \{0. 1, 2, 3\}$
$A \cap B = \{0, 1, 2, 3, 4, 5\} \cap \{0, 1, 2, 3\} = \{0. 1, 2, 3\}$
View full question & answer→Question 991 Mark
For event $A$ and $B$ of finite sample space, if $A \cap B$ are events of which type?
AnswerFor event $A$ and $B,$ if $A \cap B = \emptyset,$ then events $A$ and $B$ are called mutually exclusive events.
View full question & answer→Question 1001 Mark
Interpret the event $A \cup B$ and $A \cap B.$
AnswerEvent $A \cup B$ means the occurrence of at least one of the events $A$ and $B.$ Event $A \cap B$ means the occurrence of event $A$ and $B$ both together.
View full question & answer→