Mutually Exclusive Events

Q.

A bag contains $2$ white and $4$ black balls. A ball is drawn $5$ times with replacement. The probability that at least $4$ of the balls drawn are white is

1. $\frac{8}{141}$

2. $\frac{10}{243}$

3. $\frac{11}{243}$

4. $\frac{8}{41}$

Q.

A bag contains $6$ white and $4$ black balls. Two balls are drawn at random. The probability that they are of the same colours, is

1. $\frac{1}{15}$

2. $\frac{2}{5}$

3. $\frac{4}{15}$

4. $\frac{7}{15}$

Q.

If A, B, C are three mutually exclusive and exhaustive events of an experiment such that 3P(A) = 2P(B) = P(C), then P(A) is equal to

1. $\frac{6}{11}$

2. $\frac{1}{11}$

3. $\frac{2}{11}$

4. $\frac{5}{11}$

Q. Two dice are thrown and the sum of the numbers which come up on the dice is noted. Let us consider the following events associated with this experiment
$A:$ 'the sum is even'.
$B:$ 'the sum is a multiple of $3^{\prime }$.
$C:$ 'the sum is less than $4^{\prime }$.
$D:$ 'the sum is greater than ${11}^{\prime }$.
Which pairs of these events are mutually exclusive?

Q.

An event has odds in favor $4:5,$then the probability that occurs is

1. $\frac{1}{5}$

2. $\frac{4}{5}$

3. $\frac{4}{9}$

4. $\frac{5}{9}$

Q.

A pair of fair dice is rolled together till a sum of either $5$or $7$is obtained. Then the probability that $5$comes before $7$ is

1. $1/5$

2. $2/5$

3. $4/5$

4. None of these

Q.

If S is the sample space and P(A) = $\frac{1}{3}$ P(B) and S = $A\cup B$, where A and B are two mutually exclusive events, then P(A) =

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

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

3. $\frac{3}{4}$

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

Q.

Given two mutually exclusive events A and B such that $P\left(A\right)=\frac{1}{2}$ and $P\left(B\right)=\frac{1}{3}$ find P(A or B).

Q.

A bag contains 3 red, 4 white and 5 blue balls. All balls are different. Two balls are drawn at random. The probability that they are of different colour is

1. 1

2. $\frac{47}{66}$

3. $\frac{10}{33}$

4. $\frac{1}{3}$

Q.

A bag contains $3$ black, $3$ white and $2$ red balls. One by one, $3$ balls are drawn without replacement. The probability that the third ball is red, is

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

2. $\frac{1}{3}$

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

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

Q. Urn A contains 6 red , 4 white balls and urn B contains 4 red and 6 white balls. One ball is drawn from the urn A and placed in the urn B. Then one ball is drawn at random from urn B and placed in the urn A. if one ball is now drawn from the urn A, then the probability that it is found to be red is

1. $\frac{32}{55}$
   
2. $\frac{30}{40}$
3. $\frac{3}{4}$
4. $\frac{32}{50}$

Q.

Hannah finds purple, blue, and silver space rocks while exploring. $\frac{2}{3}$ of the rocks are purple and $\frac{3}{4}$ of the remainder are blue. If there are $3$ silver rocks, how many space rocks does Hannah have?

Q.

If $A$ and $B$ are independent events of a random experiment, such that $P(A\cap B)=\frac{1}{6}$, $P(A\cap B)=\frac{1}{3}$, then $P\left(A\right)$ is equal to

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

2. $\frac{1}{3}$

3. $\frac{5}{7}$

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

Q. A bag contains 3 red, 4 white and 5 blue balls (All balls are different). If two balls are drawn at random, then the probability that they are of different colours is:

1. $\frac{47}{66}$
2. $\frac{10}{33}$
3. $\frac{5}{22}$
4. $None$

Q.

Bag A contains 4 green and 3 red balls and bag B contains 4 red and 3 green balls. One bag is taken at random and a ball is drawn and noted it is green. The probability that it comes from bag B is

1. 2/3

2. 2/7

3. 1/3

4. 3/7

Q.

A bag has four pair of balls of four distinct colours. If four balls are picked at random (without replacement), the probability that there is atleast one pair among them have the same colour is

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

2. $\frac{8}{35}$

3. $\frac{19}{35}$

4. $\frac{27}{35}$

Q. Box $1$ contains three cards bearing numbers $1,2,3;$ box $2$ contains five cards bearing numbers $1,2,3,4,5;$ and box $3$ contains seven cards bearing numbers $1,2,3,4,5,6,7.$ A card is drawn from each of the boxes. Let $x_i$ be the number on the card drawn from the $i^{th}$ box, $i=1,2,3.$

The probability that $x_1+x_2+x_3$ is odd, is

1. $\frac{29}{105}$
2. $\frac{53}{105}$
3. $\frac{57}{105}$
4. $\frac{1}{2}$

Q.

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

1. $\frac{1}{3}\le p\le \frac{1}{2}$

2. $\frac{1}{2}<p<\frac{1}{2}$

3. $\frac{1}{2}\le p\le \frac{2}{3}$

4. $\frac{1}{2}<p<\frac{2}{3}$

Q.

The angle between the lines, Whose direction ratios are $\left(1,1,2\right)$, $\left(\sqrt{3}-1,-\sqrt{3}-1,4\right)$ is

1. ${\cos }^{-1}\left(\frac{1}{65}\right)$

2. $\frac{\mathrm{\pi }}{6}$

3. $\frac{\mathrm{\pi }}{3}$

4. $\frac{\mathrm{\pi }}{2}$

Q. A natural number x is chosen at random from the first $100$ natural numbers. The probability for the inequality $x+\frac{100}{x}>50$ to hold good is:

1. $\frac{9}{20}$
2. $\frac{14}{20}$
3. $\frac{13}{20}$
4. $\frac{11}{20}$

Q. An urn contains $5$ white and $7$ black balls. A second urn contains $7$ white and $8$ black balls. One ball is transfered from the $1^{st}$ urn to the $2^{nd}$ urn without noticing its colour. A ball is now drawn at random from the $2^{nd}$ urn. The probability that it is white is

1. $\frac{49}{192}$
2. $\frac{39}{192}$
3. $\frac{99}{192}$
4. $\frac{89}{192}$

Q.

(a) If A and B be mutualy exclusive events associated with a random experiment such that P(A) = 0.4 and P(B) = 0.5, then find :
(i) $P(A\cup B)$ (ii) $P(\overline{A}\cup \overline{B})$

(iii) $P(\overline{A}\cup B)$ (iv) $P(A\cup \overline{B})$

(b) A and B are two events such that P(A)= 0.54, P(B) = 0.69 and $P(A\cup B)$ = 0.35. Find:

(i) $P(A\cup B)$ (ii) $P(\overline{A}\cup \overline{B})$

(iii) $P(A\cup \overline{B})$ (iv) $P(B\cup \overline{A})$

(c) Fill in the blanks in the following table :

P(A) P(B)

$P(A\cap B)$ $P(A\cup B)$

(i) $\frac{1}{3}$ $\frac{1}{5}$ $\frac{1}{15}$___

(ii) 0.35 ___ 0.25 0.6

(iii) 0.5 0.35 ___ 0.7

Q. A die is thrown $6$ times. If 'getting an odd number' is a success, then the probability of at most $5$ successes is:

1. $\frac{1}{64}$
2. $\frac{63}{64}$
3. $\frac{1}{2}$
4. $\frac{1}{2^5}$

Q. Two cards are drawn from a well-shuffled pack of $52$ cards without replacement. Then the probability that one is a red queen and other is a king of black color is $p$ then the value of $663p$ is

Q.

A box contains 10 good articles and 6 with defects. One item is draqn at random. The Probability that it is either good of has a defect is

1. $\frac{64}{64}$

2. $\frac{49}{64}$

3. $\frac{40}{64}$

4. $\frac{24}{64}$

Q.

Two dice are thrown. The events A, B, C, D, E and F are described as follows :

A = Getting an even number on the first die.

B = Getting an odd number on the first die.

C = Getting at most 5 as sum of the numbers on the two dice.

D = Getting the sum of the numbers on the dice greater than 5 but less than 10.

E = Getting at least 10 as the sum of the numbers on the dice.

F = Getting an odd number on one of the dice.

(i) Describe the following events : A and B, B or C, B and C, A and E, A or F, A and F

(ii) State true or false :

(a) A and B are mutually exclusive.

(b) A and B are mutually exclusive and exhaustive events.

(c) A and C are mutually exclusive events.

(d) C and D are mutually exclusive and exhaustive events.

(e) C, D and E are mutually exclusive and exhaustive events.

(f) A' and B' are mutually exclusive events.

(g) A, B, F arc mutually exclusive and exhaustive events.

Q. The probabilities of two events A and B are 0.3 and 0.6 respectively. The probability that both A and B occur simultaneously is 0.18 Then, the Probability that neither A nor B occurs is

1. 0.1
2. 0.28
3. 0.42
4. 0.72

Q. Two dice are thrown. The events $A,B\text{and}C$ are as follows:
$A:$ getting an even number on the first die.
$B:$ getting an odd number on the first die.
$C:$ getting the sum of the numbers on the dice $\le 5$

$i.$ State true or false: (give reason for your answer)
$A$ and $B$ are mutually exclusive.
$ii.$ State true or false: (give reason for your answer)
$A$ and $B$ are mutually exclusive and exhaustive.
$iii.$ State true or false: (give reason for your answer)
$A=B\text{'}$.
$iv.$ State true or false: (give reason for your answer)
$A$ and $C$ are mutually exclusive.
$v.$ State true or false: (give reason for your answer)
$A$ and $B^{\prime }$ are mutually exclusive.
$vi.$ State true or false: (give reason for your answer)
$A\text{'},B\text{'},C$ are mutually exclusive and exhaustive.

Q.

In a swimming race $3$ swimmers compete . The probability of $A$ and $B$ winning is same and twice that of $C$.What is the probability that $B$ or $C$ wins. Assuming no two finish the race at the same time.

1. 3/5

2. 8/10

3. 2/10

4. 1/5

Q.

Three coins are tossed. Describe
(i) two events A and B which are mutually exclusive.

(ii) three events A, B and C which are mutually exclusive and exhaustive.

(iii) two events A and B which am not mutually exclusive.

(iv) two events A and B which are mutually exclusive but not exhaustive.