Question:
A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II and III are p, q and $\frac{1}{2}$ respectively. If the probability th at the student is successful, is $\frac{1}{2}$ then
A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II and III are p, q and $\frac{1}{2}$ respectively. If the probability th at the student is successful, is $\frac{1}{2}$ then
Updated On: Jun 14, 2022
- p = q = 1
- p = q = $\frac{1}{2}$
- p = 1, q = 0
- p = 1 , q = $\frac{1}{2}$
Hide Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Let A, B and C denote the events of passing the tests I,
II and III, respectively.
Evidently A, B and C are independent events.
According to given condition,
$\frac{1}{2}=P[(A \cap B)\cup (A \cap B)]$
$=P(A \cap B)+P(A \cap C)-P(A \cap B \cap C) $
= P {A) P(B) + P(A) . P(C) - P(A) . P(B) . P(C)
=pq+p $\frac{1}{2}-p\frac{1}{2}$
$\Rightarrow \, $1=2 p q + p - p q $\Rightarrow \, $ 1 = p(q + 1) $ \, \, \, \, \, \, \, \, \, \, \, \, $ ....(i)
The values of option (c) satisfy E (i).
[ Infact, E (i) is satisfied for infinite number of values of p and If we take any values of q such that $0 \le q \le 1.$
then, p takes the value $\frac{1}{q+1}$ It is evident that,
$0 < \frac{1}{q+1} \le 1 \, i.e., 0 < p \le 1 $ But we have to choose correct answer from given ones.]
II and III, respectively.
Evidently A, B and C are independent events.
According to given condition,
$\frac{1}{2}=P[(A \cap B)\cup (A \cap B)]$
$=P(A \cap B)+P(A \cap C)-P(A \cap B \cap C) $
= P {A) P(B) + P(A) . P(C) - P(A) . P(B) . P(C)
=pq+p $\frac{1}{2}-p\frac{1}{2}$
$\Rightarrow \, $1=2 p q + p - p q $\Rightarrow \, $ 1 = p(q + 1) $ \, \, \, \, \, \, \, \, \, \, \, \, $ ....(i)
The values of option (c) satisfy E (i).
[ Infact, E (i) is satisfied for infinite number of values of p and If we take any values of q such that $0 \le q \le 1.$
then, p takes the value $\frac{1}{q+1}$ It is evident that,
$0 < \frac{1}{q+1} \le 1 \, i.e., 0 < p \le 1 $ But we have to choose correct answer from given ones.]
Was this answer helpful?
0
0
Top Questions on Probability
- A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of 5. The probability that A wins if A makes the first throw is:
- JEE Main - 2025
- Mathematics
- Probability
- Bag \( B_1 \) contains 6 white and 4 blue balls, Bag \( B_2 \) contains 4 white and 6 blue balls, and Bag \( B_3 \) contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability that the ball is drawn from Bag \( B_2 \) is:
- JEE Main - 2025
- Mathematics
- Probability
- Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:
- JEE Main - 2025
- Mathematics
- Probability
- Bag \( B_1 \) contains 6 white and 4 blue balls, Bag \( B_2 \) contains 4 white and 6 blue balls, and Bag \( B_3 \) contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability that the ball is drawn from Bag \( B_2 \) is:
- JEE Main - 2025
- Mathematics
- Probability
- A die is thrown 10 times. The probability that an odd number will come up at least once is:
- KCET - 2024
- Mathematics
- Probability
View More Questions
Questions Asked in JEE Advanced exam
- Let \(S=\left\{\begin{pmatrix} 0 & 1 & c \\ 1 & a & d\\ 1 & b & e \end{pmatrix}:a,b,c,d,e\in\left\{0,1\right\}\ \text{and} |A|\in \left\{-1,1\right\}\right\}\), where |A| denotes the determinant of A. Then the number of elements in S is _______.
- JEE Advanced - 2024
- Matrices
- A positive, singly ionized atom of mass number \( A_M \) is accelerated from rest by the voltage \( 192 \, \text{V} \). Thereafter, it enters a rectangular region of width \( w \) with magnetic field \( \vec{B}_0 = 0.1\hat{k} \, \text{T} \). The ion finally hits a detector at the distance \( x \) below its starting trajectory. Which of the following option(s) is(are) correct? \[ \text{(Given: Mass of neutron/proton = } \frac{5}{3} \times 10^{-27} \, \text{kg, charge of the electron = } 1.6 \times 10^{-19} \, \text{C).} \]
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass \( m \) and radius \( r \) and is in a uniform vertical magnetic field \( B_0 \). When a current \( I \) is passed through the loop, the loop turns about the line PQ by an angle \( \theta \). The angle \( \theta \) is given by:
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A region in the form of an equilateral triangle (in x-y plane) of height \( L \) has a uniform magnetic field \( \mathbf{B} \) pointing in the \( +z \)-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height \( L \), is placed in the x-y plane with its vertex \( P \) at \( x = 0 \) in the orientation shown in the figure. At \( t = 0 \), the loop starts entering the region of the magnetic field with a uniform velocity \( \mathbf{v} \) along the \( +x \)-direction. The plane of the loop and its orientation remain unchanged throughout its motion.
Which of the following graphs best depicts the variation of the induced emf (\( \mathcal{E} \)) in the loop as a function of the distance (\( x \)) starting from \( x = 0 \)?- JEE Advanced - 2024
- Radioactivity
- A table tennis ball has radius \( \frac{3}{2} \times 10^{-2} \, \text{m} \) and mass \( \frac{22}{7} \times 10^{-3} \, \text{kg} \). It is slowly pushed down into a swimming pool to a depth \( d = 0.7 \, \text{m} \) below the water surface and then released from rest. It emerges from the water surface at speed \( v \), without getting wet, and rises to a height \( H \). Which of the following option(s) is(are) correct?\(\text{(Given: } \pi = \frac{22}{7}, g = 10 \, \text{m/s}^2, \text{density of water} = 1 \times 10^3 \, \text{kg/m}^3, \text{viscosity of water} = 1 \times 10^{-3} \, \text{Pa.s).}\)
- JEE Advanced - 2024
- Radioactivity
View More Questions
Concepts Used:
Probability
Probability is defined as the extent to which an event is likely to happen. It is measured by the ratio of the favorable outcome to the total number of possible outcomes.
The definitions of some important terms related to probability are given below:
Sample space
The set of possible results or outcomes in a trial is referred to as the sample space. For instance, when we flip a coin, the possible outcomes are heads or tails. On the other hand, when we roll a single die, the possible outcomes are 1, 2, 3, 4, 5, 6.
Sample point
In a sample space, a sample point is one of the possible results. For instance, when using a deck of cards, as an outcome, a sample point would be the ace of spades or the queen of hearts.
Experiment
When the results of a series of actions are always uncertain, this is referred to as a trial or an experiment. For Instance, choosing a card from a deck, tossing a coin, or rolling a die, the results are uncertain.
Event
An event is a single outcome that happens as a result of a trial or experiment. For instance, getting a three on a die or an eight of clubs when selecting a card from a deck are happenings of certain events.
Outcome
A possible outcome of a trial or experiment is referred to as a result of an outcome. For instance, tossing a coin could result in heads or tails. Here the possible outcomes are heads or tails. While the possible outcomes of dice thrown are 1, 2, 3, 4, 5, or 6.