Mutually Exclusive Events

Q.

The probabilities of three mutually exclusive events are $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ and $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$. The statements is

1. True

2. Wrong

3. Could be either

4. Do not know

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

1. $\frac{1}{4}$
2. $\frac{1}{5}$
3. $\frac{1}{2}$
4. $\frac{1}{3}$

Q. Which of the following pair of events are "mutually exclusive" ?

1. Getting a head or a tail in a toss of a coin.
2. Getting a '1' or getting a '6' in a single throw of a die.
3. Drawing a 'king' or drawing a 'red card' from a pack of 52 cards.
4. Drawing a 'red card' or drawing a 'spade' from a pack of 52 cards.

Q. Which of the following pair of events are non-mutually exclusive events in a single throw of a die?

1. Getting a perfect cube number or getting an odd number.
2. Getting an even number or getting a perfect square number.
3. Getting an odd number or getting an even number.
4. Getting a number divisible by '1' or getting a number divisible by '2'.

Q. All the possible squares are made using combination of smaller squares in the grid. What is the probability of picking a square that is formed by at least two different-colored squares?

1. 54/91
2. 37/91
3. 5/13
4. 1/13

Q. Three dice are rolled simultaneously. The probability of getting the total sum of more than $5$ is

1. $\frac{5}{54}$
2. $\frac{67}{72}$
3. $\frac{103}{108}$
4. $\frac{53}{54}$

Q. Write sample space ‘S’ and number of sample point n(S) for each of the following experiments. Also write events A, B, C in the set form and write n(A), n(B), n(C).
(1) One die is rolled,
Event A : Even number on the upper face.
Event B : Odd number on the upper face.
Event C : Prime number on the upper face.

(2) Two dice are rolled simultaneously,
Event A : The sum of the digits on upper faces is a multiple of 6.
Event B : The sum of the digits on the upper faces is minimum 10.
Event C : The same digit on both the upper faces.

(3) Three coins are tossed simultaneously.
Condition for event A : To get at least two heads.
Condition for event B : To get no head.
Condition for event C : To get head on the second coin.

(4) Two digit numbers are formed using digits 0, 1, 2, 3, 4, 5 without repetition of the digits.
Condition for event A : The number formed is even
Condition for event B : The number formed is divisible by 3.
Condition for event C : The number formed is greater than 50.

(5) From three men and two women, environment committee of two persons is to be formed.
Condition for event A : There must be at least one woman member.
Condition for event B : One man, one woman committee to be formed.
Condition for event C : There should not be a woman member.

(6) One coin and one die are thrown simultaneously.
Condition for event A : To get head and an odd number.
Condition for event B : To get a head or tail and an even number.
Condition for event C : Number on the upper face is greater than 7 and tail on the coin.

Q. Match the following events with their corresponding mutually exclusive events on rolling of a die.

1. An odd number
2. Even prime number
3. Getting '4' or '1'
4. Getting '3' or '4'

Q. Which of the following statement(s) is/are true for the function
$f\left(x\right)=(x-1{)}^2(x-2)+1$ defined on $[0,2]$?

1. Range of $f$ is $\left[\frac{23}{27},1\right]$
2. The coordinates of the turning point of the graph of $y=f\left(x\right)$ occurs at $(1,1)$ and $\left(\frac{5}{3},\frac{23}{27}\right).$
3. The value of $p$ for which the equation $f\left(x\right)=p$ has $3$ distinct solutions lies in interval $\left(\frac{23}{27},1\right).$
4. The area enclosed by $y=f\left(x\right),$ the lines $x=0$ and $y=1$ as $x$ varies from $0$ to $1$ is $\frac{7}{12}.$

Q. A die is thrown then find the probability of getting
(i) an odd number.
(ii) a perfect square.
(iii) a number greater than 3.

Q.

Events A and B are such that P(not A or not B) = 0.44. State whether A and B are mutually exclusive events. (Yes/No)

1. No

2. Yes

Q. A rifle man firing at a distant target and has only 10% chance of hitting it. The minimum number of rounds he must fire in order to have 50% chance of hitting it atleast once is

1. 6
2. 7
3. 9
4. 8

Q. $\text{Which of the following pair of}$
$\text{events are mutually exclusive events}$
$\text{on rolling a dice once}?$

1. $\text{Getting an even number}$
   $\text{or getting a perfect number.}$
2. $\text{Getting an even number}$
   $\text{or getting a perfect square number.}$
3. $\text{Getting an odd number}$
   $\text{or getting a perfect number}$
4. $\text{Getting an even prime number}$
   $\text{or getting a perfect number.}$

Q. In a swimming race 3 swimmers compete . The probability of A and B wining 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. $\frac{2}{10}$
2. $\frac{1}{5}$
3. $\frac{3}{5}$
4. $\frac{8}{10}$

Q. In a class test in English $10$ students scored $75$ marks, $12$ students scored $60$ marks, $8$ scored $40$ marks and $3$ scored $30$ marks, the mode for their score is

1. $75$
2. $30$
3. $60$
4. $25$

Q. An experiment has $10$ equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is

1. $4$ or $8$
2. $3,6$ or $9$
3. $2,4$ or $8$
4. $5$ or $10$

Q. In each of the following experiments, write the sample space S, number of sample points n(S), event A, B, C and n(A), n(B), n(C), also find complementary events, mutually exclusive events:

(ii) A die is thrown. A is the event that prime number comes up, B is the event that the number is divisible by three comes up, C is the event that the perfect square number comes up .

Q.

Given that N = {1, 2, 3, 4, …… 50} . Then write

a) the subset of N consisting of all the prime numbers.

b) the subset of N consisting of all the perfect cubes.

Q. An experiment has $24$ equally likely outcome . let A and B two non-empty events of the experiments . If B consists of $9$ outcome the number of outcomes that A must have so that A and B are independent events is

1. $8$ or $16$ or $24$
2. $2$ or $4$ or $8$
3. $3$ or $6$ or $9$
4. $4$ or $8$ or $12$

Q.

A drawer contains $5$ brown and $4$ blue socks well mixed. A man reaches the drawer and pulls out $2$ socks at random. What is the probability that they match ?

1. $\frac{2}{9}$

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

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

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

Q. $\text{A letter is choosen at random from the}$
$\text{word 'MATH'. The following are the}$
$\text{events associated wth the experiment.}$
$E_1:\text{Selecting a vowel}$
$E_2:\text{Selecting a consonant}$
$E_3:\text{Selecting letters 'A' and 'H'}$
$E_4:\text{Selecting letters 'M' and 'T'}$
$\text{Which of the following pair of}$
$\text{events are mutually exclusive}$
$\text{events?}$

1. $E_1,E_2$
2. $E_1,E_3$
3. $E_2,E_4$
4. $E_1,E_4$

Q. A die is thrown:
P is the event of getting an odd number.
Q is the event of getting an even number.
R is the event of getting a prime number.

Q.

A die is rolled and two events A and B are defined as follows.
A: An odd number turns up
B: A prime Number turns up

Find the value of $36P(A\cup B)$.

___

Q. Which rational expression should be subtracted from $\frac{x^2+1}{x-1}$ to get $\frac{x-3}{x+1}$?

Q. Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday). Each is equally likely to visit the shop on any day as on another day. What is the probability that both will visit the shop on (i) the same day? (ii) consecutive days? (iii) different days?

Q. Jia is learning about the expanded form of numbers.
Her teacher asks her to write a random five-digit number and write its expanded form.
Jia wrote the number 49638.
Help Jia select the correct expanded form for the number 49638.

1. $4\times 10000+9\times 1000+3\times 100$ $+6\times 10+8\times 1$
2. $4\times 10000+9\times 100+6\times 100$ $+3\times 10+8\times 1$
3. $4\times 10000+9\times 1000+6\times 100$ $+3\times 10+8\times 1$
4. $4\times 1000+9\times 100+6\times 10$ $+3\times 10+8\times 1$