Simple and Compound Event

Q.

A bag contains 4 white and 5 black balls. Another bag contains 9 white and 7 black balls. A ball is transferred from the first bag to the second and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.

Q.

An experiment consists of flipping a coin and then flipping it a second time if head occurs. if a tail occurs on the first flip, then a six-faced die is tossed once. assuming that the outcomes are equally likely, what is the probability of getting one head and tail?

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

2. $\frac{1}{36}$

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

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

Q. If the letters of the word MATHEMATICS are arranged arbitrarily, the probability that C comes before E, E before H, H before I and I before S is

1. $\frac{1}{75}$
2. $\frac{1}{24}$
3. $\frac{1}{120}$
4. $\frac{1}{720}$

Q.

An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a balls is drawn at random. What is the probability that the second ball is red?

Q. A bag contains 5 white and 5 black balls. Second bag contains 8 white and 6 black balls. A ball is transferred from the first bag to the second bag. A ball is drawn randomly from the second bag. Find the probability that the ball drawn is white.

Q. A slip of paper is given to a person A who marks it either with a plus sign or a minus sign. The probability of his writing a plus sign is $\frac{1}{3}$. A passes the slip to B, who may either leave it alone or change the sign before passing it to C. Next C passes the slip to D after perhaps changing the sign. Finally D passes it to a referee after perhaps changing the sign. B, C, D each change the sign with probability $\frac{2}{3}$.
If the referee observes a plus sign on the slip then the probability that A originally wrote a plus sign is

1. $\frac{14}{27}$
2. $\frac{13}{41}$
3. $\frac{27}{14}$
4. $\frac{13}{27}$

Q.

Three dice are thrown. the probability of getting a sum that is a perfect square is:

Q.

Two natural numbers are chosen at random from the first one hundred natural numbers. The probability that the product of the chosen numbers is a multiple of 7 is.

1. $\frac{{14}_{C_2}}{{100}_{C_2}}+\frac{7_{C_1}{14}_{C_1}}{{100}_{C_2}}$

2. $\frac{{14}_{C_1}\times {13}_{C_1}}{{100}_{C_2}}$

3. $\frac{{14}_{C_1}{86}_{C_1}}{{100}_{C_2}}+\frac{{14}_{C_2}}{{100}_{C_2}}$

4. $\frac{{14}_{C_2}}{{100}_{C_2}}+\frac{{14}_{C_1}{13}_{C_1}}{{100}_{C_2}}$

Q. There are only two women among 20 persons taking part in a pleasure trip. The 20 persons are divided into two groups, each group consisting of 10 persons. Then the probability that the two women will be in the same group is

1. 
2. 
3. 
4. none

Q.

Three coins are tossed once. Describe the following events associated with this random experiment :

A = Getting three heads,

B = Getting two heads and one tail,

C = Getting three tails,

D = Getting a head on the first coin.

(i) Which pairs of events are mutually exclusive?

(ii) Which events arc elementary events?

(iii) Which events are compound events?

Q.

Three letters, each of which corresponds to an envelope, are placed in the envelopes at random.

The probability that all the letters are not placed in the right envelopes, is

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

2. $\frac{5}{6}$

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

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

Q.

Determine $P\left(\frac{E}{F}\right)$
Two coins are tossed once, where
E: Tail appears on one coin F: one coin shows head

E: no tail appears F : no head appears

Q. Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of the random variable X, and hence find the mean of the distribution.

Q. There are $n$ white and $n+1$ black balls in urn $A$, there are $n+1$ white balls and $n$ black balls in urn $B$. One ball is drawn from urn $A$ and put into urn $B$. Then two balls are drawn from urn $B$ and put into urn $A$. When the operation is completed, the probability that urn $A$ contains same number of white and black balls is $\frac{13}{25}$. Then the number of balls in urn $A$ at the start of the operation was

Q.

If two balanced dice are tossed once, the probability of the event, that the sum of the integers coming on the upper sides of the two dice is 9, is [MP PET 1987]

1. 

2. 

3. 

4. 

Q. Three coins are tossed
$i.$ Describe two events which are mutually exclusive.
$ii.$ Describe three events which are mutually exclusive and exhaustive.
$iii.$ Describe two events, which are not mutually exclusive.
$iv.$ Describe two events which are mutually exclusive but not exhaustive.
$v.$ Describe three events which are mutually exclusive but not exhaustive.

Q.

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Poonam flipped a fair coin five times. In the first three flips, the coins came up heads exactly twice and in the last three}& \text{(P)}& \frac{3}{4}\\ & \text{flips, the coin also came up heads exactly twice. The probability that the third flip was head, is}& & \\ \text{(B)}& \text{A box contains 10 transistors of which 2 are defective. Transistors are drawn one by one without replacement unless a}& \text{(Q)}& \frac{2}{5}\\ & \text{non-defective one is chosen. The probability that atmost 3 transistors are drawn, is}& & \\ & \text{A box contains 1 black and 1 white ball. A ball is drawn randomly and replaced in the box with an additional ball of the}& & \\ \text{(C)}& \text{same colour, then a second ball is drawn randomly from the box containing 3 balls. The probability that the first drawn}& \text{(R)}& \frac{4}{5}\\ & \text{ball was white given that at least one of the two balls drawn was white, is}& & \\ \text{(D)}& \text{Let}P\left(A\right)=0.7\text{and}P\left(B\right)=\mathrm{0.3.}\text{Let}{B}^c\text{denote the complement the event}B.\text{Then the smallest value of}P(A\cap B^c)\text{is}& \text{(S)}& \frac{2}{3}\\ & & \text{(T)}& \text{None of these}\end{array}$

Which of the following is a CORRECT combination?

1. $\text{(C)}\to \text{(P)},\text{(D)}\to \text{(Q)}$
2. $\text{(C)}\to \text{(S)},\text{(D)}\to \text{(Q)}$
3. $\text{(C)}\to \text{(R)},\text{(D)}\to \text{(T)}$
4. $\text{(C)}\to \text{(Q)},\text{(D)}\to \text{(S)}$

Q. A coin is tossed two times, what is the probability of getting head at least once.

Q. There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tails 40% of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?

Q. The number of ways in which we can get a sum of $11$ by throwing three dice is :

1. $56$
2. $8$
3. $27$
4. $45$

Q.

The lottery box contains tickets numbered 1-10. 2 tickets are drawn at random without replacement. The probability that the difference between the numbers on the ticket $>4$ is ?

1. 13/30

2. 15/30

3. 7/30

4. 14/30

Q.

At a fete cards bearing numbers $1$ to $500$, one on each card, are put in a box. Each player selects one card at random and that card is not replaced. If the selected card bears a number which is a perfect square of an even number the player wins prize. What is the probability that the first player wins a prize$?$

Q.

Which of the following cannot be valid assignment of probability for elementary events or outcomes of sample space

$S=\{w_1,w_2,w_3,w_4,w_5,w_6,w_7\}$:
Elementary events :

(i) $\begin{array}{ccccccc}{w}_1& w_2& w_3& w_4& w_5& w_6& w_7\\ 0.1& 0.01& 0.05& 0.03& 0.01& 0.2& 0.6\end{array}$

(ii) $\begin{array}{ccccccc}\frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}\end{array}$

(iii) $\begin{array}{ccccccc}0.7& 0.6& 0.5& 0.4& 0.3& 0.2& 0.1\end{array}$

(iv) $\begin{array}{ccccccc}\frac{1}{14}& \frac{1}{14}& \frac{1}{14}& \frac{1}{14}& \frac{1}{14}& \frac{1}{14}& \frac{1}{14}\end{array}$

Q. A bag contain $30$ tickets numbered from $1,2,3,\dots ,30$ of which four are drawn at random and arranged in ascending order $(t_1<t_2<t_3<t_4)$. Then the probability of $t_3$ being $18$ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

1. $\frac{324}{6289}$
2. $\frac{544}{9135}$
3. $\frac{414}{9139}$
4. $\frac{613}{9372}$

Q.

Cards with number $2$ to $101$ are placed in a hat. Find the probability of selecting a square number.

Q.

A number cube is rolled $360$ times and the results are recorded as follows: $41$ ones, $54$ twos, $62$ threes, $75$ fours, $33$ fives, and $95$ sixes. What is the experimental probability of rolling a two or a three?

1. $0.32$

2. $0.18$

3. $0.07$

4. $0.68$

Q. Two dice are tossed once. The probability of getting an even number at the first die or a total of $8$ is

1. $\frac{1}{36}$
2. $\frac{3}{36}$
3. $\frac{11}{36}$
4. $\frac{20}{36}$

Q.

Three coins are tossed once. Let A denote the event ‘three heads show”, B denote the event “two heads and one tail show”. C denote the event “three tails show” and D denote the event ‘a head shows on the first coin”. Which events are

(i) mutually exclusive? (ii) simple? (iii) compound?

Q. If a $3$-digit number is randomly chosen, what is the probability that either the number itself or some permutation of the number (which is a $3$-digit number) is divisible by $4$ and $5$?

1. $\frac{1}{45}$
2. $\frac{29}{180}$
3. $\frac{11}{60}$
4. $\frac{1}{4}$

Q. Toss three fair coins simultaneously and record the outcomes. Find the probability of getting atmost one head in the three tosses.

1. $\frac{1}{6}$
2. $\frac{1}{4}$
3. $\frac{1}{2}$
4. $\frac{1}{3}$