Exhaustive Events

Q. The probability that in a family of $4$ children there will be atleast one boy is

1. $\frac{13}{16}$
2. $\frac{15}{16}$
3. $\frac{14}{16}$
4. $\frac{12}{16}$

Q.

If $A$ and $B$ are two independent events such that $P(A\cup B)=0.8$ and $P\left(A\right)=0.3$. Then, $P\left(B\right)$ is equal to

1. $\frac{2}{7}$

2. $\frac{2}{3}$

3. $\frac{3}{8}$

4. $\frac{1}{8}$

Q. Next year there will be three candidates Mr.X, Mr.Y and Mr.Z for the position of a principal of a degree college exclusively meant for boys. Their respective chances for the selection are in proportion $4:2:3$. The probabilities that these persons, if selected will introduce co-education in the college, are respectively $0.3,0.5$ and $0.8$. The probability, that there will be co-education in the college, next year is $\frac{23}{9K}$, then $K$ is

Q. Can be 2 non mutually exclusive events independent?

Q.

For the two events $A$and$B,P\left(A\right)=0.38,P\left(B\right)=0.41$, then the value of $P\left(\overline{A}\right)$ is

1. $0.41$

2. $0.62$

3. $0.59$

4. $0.21$

Q.

A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16. Determine (i) P(not A), (ii) P (not B) and (iii) P(A or B).

Q. Match the following

If $E_1$ and $E_2$ are the mutually exclusive events	A	i	$E_1\cap E_2=E_1$
If $E_1$ and $E_2$ are the mutually exclusive and exhaustive events	B	ii	$(E_1-E_2)\cup (E_1\cap E_2)=E_1$
If $E_1$ and $E_2$ have common outcomes, then	C	iii	$E_1\cap E_2=\varphi ,E_1\cup E_2=S$
If $E_1$ and $E_2$ are two events such that $E_1\subset E_2$	D	iv	$E_1\cap E_2=\varphi$

Q. If $E_1$ and $E_2$ are two events such that $P\left(E_1\right)=\frac{1}{4},P\left(\frac{E_2}{E_1}\right)=\frac{1}{2}$ and $P\left(\frac{E_1}{E_2}\right)=\frac{1}{4},$ then which of the following is/are correct?

1. $E_1$ and $E_2$ are mutually independent events
2. $E_1$ and $E_2$ are exhaustive events
3. $E_2$ is twice as likely to occur as $E_1$
4. $P(E_1\cap E_2),P\left(E_1\right),P\left(E_2\right)$ are in G.P

Q. In venn diagram if two events are different (independent) then how intersection is not zero[p(anb)]

Q. From $10$ cards numbered $1$ to $10,$ a card is selected at random. consider the following events:
$A:$ The number on the card is a perfect square.
$B:$ The number is divisible by $3.$
$C:$ The number on the card is even.
$D:$ The number on the card is odd.
If $S$ is the sample space, then which of the following statement(s) is/are correct ?

1. $A^c\cup B^c=S$
2. $(A^c\cup B^c)\cup (A\cap B)=S$
3. $C^c=D$
4. $D^c=C$

Q.

Given two independent events A and B such that P (A) = 0.3, P (B) = 0.6. Find

(i) P (A and B) (ii) P (A and not B)

(iii) P (A or B) (iv) P (neither A nor B)

Q. If $A,B$ and $C$ are mutually exclusive and exhaustive events, then $P\left(A\right)+P\left(B\right)+P\left(C\right)$ equals to -

1. $1$
2. Any value between $0$ and $1$
3. $\frac{1}{3}$
4. $0$

Q. If $\frac{(1+3p)}{3},\frac{(1-p)}{4}$ and $\frac{(1-2p)}{2}$ are the probabilities of three mutually exclusive events, then the set of all values of p is

1. 
2. 
3. 
4. 

Q. If A and B are two independent events, then write P (A ∩ $\overline{B}$) in terms of P (A) and P (B).

Q. If A and B are two independent events such that P ($\overline{A}$ ∩ B) = 2/15 and P (A ∩ $\overline{B}$) = 1/6, then find P (B).

Q.

A series with three matches between India and South Africa is about to start. Consider the three events associated with the result of the match.
A. SA wins at least two matches
B. SA wins exactly one match
C. SA wins no match

Statement: The three events A, B and C are mutually exclusive and exhaustive events.

1. False

2. True

Q. Total numbers that can be made using the digits $3,4,5,6,7$ and $8$ lying between $3000$ and $4000$, which are divisible by $5$ while repetition digits is not allowed is

1. $12$
2. $24$
3. $60$
4. $120$

Q.

If, find P (A ∩ B) if A and B are independent events.

Q. An event in which all the possible outcomes of the experiments are present is known as ______ event.

1. a complement.
2. an experiment.
3. a sample space.
4. an exhaustive event.

Q.

Let A and B be independent events with P (A) = 0.3 and P (B) = 0.4. Find

(i) P (A ∩ B) (ii) P (A ∪ B)

(iii) P (A|B) (iv) P (B|A)

Q. If A, B, C are mutually exclusive and exhaustive events associated to a random experiment, then write the value of P (A) + P (B) + P (C).

Q. Events $E_1,E_2,....E_n$ are ______ events if their union is the sample space $S$.

1. complement
2. exhaustive
3. impossible event
4. sure event

Q.

Check whether the following probabilities P(A) and P(B) are consistently defined

(i) P(A) = 0.5, P(B) = 0.7, P(A ∩ B) = 0.6

(ii) P(A) = 0.5, P(B) = 0.4, P(A ∪ B) = 0.8

Q. Given two independent events A and B such that $P\left(A\right)=0.3$, $P\left(B\right)=0.6$. Find
(i) P(A and not B) (ii) P(neither A nor B)

Q. Mark the correct alternative in the following question:

$$\begin{gathered} \mathrm{If}\mathrm{the}\mathrm{events}A\mathrm{and}B\mathrm{are}\mathrm{independent},\mathrm{then}\mathrm{P}\left(A\cap B\right)\mathrm{is}\mathrm{equal}\mathrm{to} \\ \left(\mathrm{a}\right)\mathrm{P}\left(A\right)+\mathrm{P}\left(B\right)\left(\mathrm{b}\right)\mathrm{P}\left(A\right)-\mathrm{P}\left(B\right)\left(\mathrm{c}\right)\mathrm{P}\left(A\right)\mathrm{P}\left(B\right)\left(\mathrm{d}\right)\frac{\mathrm{P}\left(A\right)}{\mathrm{P}\left(B\right)} \end{gathered}$$

Q. If $\frac{1+4p}{4},\frac{1-p}{4},\frac{1-2p}{4}$ are probabilities of three mutually exclusive and exclusive and exhaustive events, then the possible value of p belong to the set

1. $(0,\frac{2}{3})$
2. $[0,\frac{1}{2}]$
3. $[-\frac{1}{4},\frac{1}{2}]$
4. $[-\frac{2}{3},\frac{2}{3}]$

Q. A batsman is going to face a delivery. Following events are described.
X = {He will get out}
Y = {He will score one or more runs on this delivery}
Z = {He will not score any run on this delivery}
Events X , Y and Z are exhaustive events.

1. True
2. False

Q. A sample space S is given as S = {1, 2, 3, 4, 5, 6, 7}
Following events are described for this sample space.
Event X = {1, 2, 3}
Y = {3, 4, 5}
Z = {5, 6, }
Events X , Y and Z are not exhaustive events because event X and event Y have an element in common

1. True
2. False

Q. Hi,
this is from AKSHAY
The question is as follows,
A series with three matches between India and South Africa is about to start. Consider the three events associated with the result of the match.
A.SA wins at least two matches
B.SA wins exactly one match
C.SA wins no match
Statement: The three events A, B and C are mutually exclusive and exhaustive events.

I can understand the answer but except last statement that
A_{​​​​​}​​​​​​ Intersection B is null , B intersection C is null , C intersection A is null
How it comes?

Q. Consider the following statements:
1. If $A$ and $B$ are exhaustive events, then their union is the sample space.
2. If $A$ and $B$ are exhaustive events, then their intersection must be an empty event.
Which of the above statements is/are correct?

1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2