If A and B are two independent events with $\text{P(A)}=\frac{1}{3}$ and $\text{P(B)}=\frac{1}{4},$ then P(B'|A) is equal to:
- A$\frac{1}{4}$
- B$\frac{1}{3}$
- C$\frac{3}{4}$
- D$1$
Question types
Probability question types
377 questions across 7 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.
377
Questions
7
Question groups
5
Question types
01
M.C.Q (1 Marks)
220 Q→02Assertion (A) & Reason (B) MCQ
10 Q→031 Marks
77 Q→042 Marks
25 Q→053 Marks
7 Q→064 Marks
5 Q→07Case study (4 Marks)
33 Q→Sample Questions
Probability questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The probability distribution of a random variable $X$ is:
where $k$ is some unknown constant.
The probability that the random variable $X$ takes the value 2 is:
| X | 0 | 1 | 2 | 3 | 4 |
| P(X) | 0.1 | k | 2k | k | 0.1 |
The probability that the random variable $X$ takes the value 2 is:
- A$\frac{1}{5}$
- B$\frac{2}{5}$
- C$\frac{4}{5}$
- D1
Let $E$ be an event of a sample space $S$ of an experiment then $P(S \mid E)=$
- A$P(S \cap E)$
- B$P(E)$
- C1
- D$0$
If $A$ and $B$ are events such that $P(A / B)=P(B / A) \neq 0$, then
- A$A \subset B$, but $A \neq B$
- B$A=B$
- C$A \cap B=0$
- D$P(A)=P(B)
A problem in Mathematics is given to three students whose chances of solving it are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is
- A$\frac{1}{4}$
- B$\frac{1}{3}$
- C$\frac{1}{2}$
- D$\frac{3}{4}$
Assertion (A): Two coins are tossed simultaneously. The probability of getting two heads, if it is known that at least one head comes up, is $\frac{1}{3}$.
Reason $(R)$ : Let $E$ and $F$ be two events with a random experiment, then $P(F / E)=\frac{P(E \cap F)}{P(E)}$.
Reason $(R)$ : Let $E$ and $F$ be two events with a random experiment, then $P(F / E)=\frac{P(E \cap F)}{P(E)}$.
- ABoth Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
- BBoth Assertion (A) and Reason (R) are true, but Reason $(R)$ is not the correct explanation of the Assertion (A).
- CAssertion (A) is true and Reason (R) is False.
- DAssertion (A) is false but Reason (R) is true.
A man $P$ speaks truth with probability $p$ and an other $\operatorname{man} Q$ speaks truth with probability $2 p$.
Assertion (A) : If $P$ and $Q$ contradict each other with probability $1 / 2$, then there are two values of $p$.
Reason (R) : A quadratic equation with real coefficients has two real roots.
Assertion (A) : If $P$ and $Q$ contradict each other with probability $1 / 2$, then there are two values of $p$.
Reason (R) : A quadratic equation with real coefficients has two real roots.
- ABoth (A) and (R) are true and (R) is the correct explanation of (A).
- BBoth (A) and (R) are true but (R) is not the correct explanation of (A).
- ✓(A) is true but (R) is false.
- D(A) is false but (R) is true.
Answer: C.
View full solution →Assertion (A) : 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 ball is drawn at random. Then, the probability that the second ball is red is $\frac{1}{2}$.
Reason (R) : By Bayes' Theorem
$
\begin{array}{r}
P\left(E_1 \mid E\right)=\frac{P\left(E \mid E_1\right) P\left(E_1\right)}{P\left(E_1\right) P\left(E \mid E_1\right)+P\left(E_2\right) P\left(E \mid E_2\right)+P\left(E_3\right) P\left(E \mid E_3\right)}
\end{array}
$
Reason (R) : By Bayes' Theorem
$
\begin{array}{r}
P\left(E_1 \mid E\right)=\frac{P\left(E \mid E_1\right) P\left(E_1\right)}{P\left(E_1\right) P\left(E \mid E_1\right)+P\left(E_2\right) P\left(E \mid E_2\right)+P\left(E_3\right) P\left(E \mid E_3\right)}
\end{array}
$
- ✓Both (A) and (R) are true and (R) is the correct explanation of (A).
- BBoth (A) and (R) are true but (R) is not the correct explanation of (A).
- C(A) is true but (R) is false.
- D(A) is false but (R) is true.
Answer: A.
View full solution →Assertion (A) : Bag I contains 3 red and 4 black balls while another bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Then, the probability that it was drawn from bag II is $\frac{35}{68}$.
Reason (R) : By Bayes' theorem,
$
\begin{array}{r}
P\left(\frac{E_1}{A}\right)=\frac{P\left(E_1\right) \cdot P\left(A / E_1\right)}{P\left(E_1\right)\left(A \mid E_1\right)+P\left(E_1\right) \cdot P\left(A \mid E_2\right)+P\left(E_1\right) P\left(A \mid E_3\right)}
\end{array}
$
Reason (R) : By Bayes' theorem,
$
\begin{array}{r}
P\left(\frac{E_1}{A}\right)=\frac{P\left(E_1\right) \cdot P\left(A / E_1\right)}{P\left(E_1\right)\left(A \mid E_1\right)+P\left(E_1\right) \cdot P\left(A \mid E_2\right)+P\left(E_1\right) P\left(A \mid E_3\right)}
\end{array}
$
- ABoth (A) and (R) are true and (R) is the correct explanation of (A).
- BBoth (A) and (R) are true but (R) is not the correct explanation of (A).
- ✓(A) is true but (R) is false.
- D(A) is false but (R) is true.
Answer: C.
View full solution →Assertion (A) : Let $E$ and $F$ be events associated with the sample space $S$ of an experiment. Then, we have $P(S \mid F)=P(F \mid F)=1$.
Reason (R) : If $A$ and $B$ are any two events associated with the sample space $S$ and $F$ is an event associated with $S$ such that $P(F) \neq 0$, then $P((A \cup B) \mid F)=P(A \mid F)+P(B \mid F)-P((A \cap B) \mid F)$
Reason (R) : If $A$ and $B$ are any two events associated with the sample space $S$ and $F$ is an event associated with $S$ such that $P(F) \neq 0$, then $P((A \cup B) \mid F)=P(A \mid F)+P(B \mid F)-P((A \cap B) \mid F)$
- ABoth (A) and (R) are true and (R) is the correct explanation of (A).
- ✓Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C(A) is true but (R) is false.
- D(A) is false but (R) is true.
Answer: B.
View full solution →An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known:
P(A fails) = 0.2
P(B fails alone) = 0.15
P(A and B fail) = 0.15
Evaluate the following probabilities P(A fails alone).
P(A fails) = 0.2
P(B fails alone) = 0.15
P(A and B fail) = 0.15
Evaluate the following probabilities P(A fails alone).
An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known:
P(A fails) = 0.2
P(B fails alone) = 0.15
P(A and B fail) = 0.15
Evaluate the following probabilities P(A fails|B has failed).
P(A fails) = 0.2
P(B fails alone) = 0.15
P(A and B fail) = 0.15
Evaluate the following probabilities P(A fails|B has failed).
If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability $\frac{1}{2}$).
View full solution →If a leap year is selected at random, what is the chance that it will contain 53 tuesdays?
If A and B are any two events such that P(A) + P(B) - P(A and B) = P(A), then
Suppose that 90% of people are right-handed. What is the probability that at most of 6 of a random sample of 10 people are right-handed?
A couple has two children, find the probability that both children are females, if it is known that the elder child is a female.
A couple has two children, find the probability that both children are males, if it is known that at least one of the children is male.
A and B are two events such that P (A) $\ne$ 0. Find P(B|A), if A is a subset of B.
View full solution →Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability if 0.3, if the second group wins. Find the probability that the new product introduced was by the second group.
Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.
Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.
Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain. A grade and 20% of day scholars attain A grade in their annual examination. At the end of year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostlier?
Probability that A speaks truth is $\frac{4}{5}$. A coin is tossed. A reports that a head appears. The probability that actually there was head is
View full solution →Suppose we have four boxes A, B, C and D containing coloured marbles as given below:
| Box | Marble colour | ||
|
| Red | White | Black |
| A B C D | 1 6 8 0 | 6 2 1 6 | 3 2 1 4 |
One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A?, box B? box C?
A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i. e if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?
A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.
A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.
A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?
Akash and Prakash appeared for first round of an interview for two vacancies. The probability of Nisha's selection is $\frac{1}{3}$ and that of Ayushi's selection is $\frac{1}{2}$.
View full solution →
(i) Find the probability that both of them are selected.
(ii) The probability that none of them is selected.
In an office three employees Govind, Priyanka and Tahseen process incoming copies of a certain form. Govind process $50 \%$ of the forms, Priyanka processes $20 \%$ and Tahseen the remaining $30 \%$ of the forms. Govind has an error rate of 0.06 , Priyanka has an error rate of 0.04 and Tahseen has an error rate of 0.03 .
View full solution →
(i) The manager of the company wants to do a quality check. During inspection he selects a form at random from the days output of processed forms. If the form selected at random has an error, find the probability that the form is NOT processed by Govind.
(ii) Find the probability that Priyanka processed the form and committed an error.
Shama is studying in class XII. She wants do graduate in chemical engineering. Her main subjects are mathematics, physics, and chemistry. In the examination, her probabilities of getting grade A in these subjects are $0.2,0.3$, and 0.5 respectively.
View full solution →
(i) Find the probability that she gets grade A in all subjects.
(ii) Find the probability that she gets grade A in no subjects.
Family photography is all about capturing groups of people that have family ties. These range from the small group, such as parents and their children. New-born photography also falls under this umbrella. Mr Ramesh, His wife Mrs Saroj, their daughter Sonu and son Ashish line up at random for a family photograph, as shown in figure.
(i) Find the probability that daughter is at one end, given that father and mother are in the middle.
(ii) Find the probability that mother is at right end, given that son and daughter are together.
Mr. Ajay is taking up subjects of mathematics, physics, and chemistry in the examination. His probabilities of getting a grade $\mathrm{A}$ in these subjects are $0.2,0.3$, and 0.5 respectively.
View full solution →
(i) Find the probability that Ajay gets Grade A in all subjects.
(ii) Find the probability that he gets Grade A in no subjects.
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