A die is rolled. Consider events : \( A = \{1, 2, 5\} \), \( B = \{3, 5\} \), \( C = \{2, 3, 4, 5\} \). Find \( P(A \cap B|C) \) and \( P(A \cup B|C) \).
Show Hint
Solution and Explanation
• \( P(X|C) = \frac{n(X \cap C)}{n(C)} \).
• Set operations: Intersection \( \cap \) (common elements), Union \( \cup \) (all elements from both).
Step 1: Find \( P(A \cap B|C) \)
First, find \( A \cap B = \{5\} \).
Now find the intersection with \( C \): \( (A \cap B) \cap C = \{5\} \cap \{2, 3, 4, 5\} = \{5\} \).
\[ P(A \cap B|C) = \frac{n((A \cap B) \cap C)}{n(C)} = \frac{1}{4} \]
Step 2: Find \( P(A \cup B|C) \)
First, find \( A \cup B = \{1, 2, 3, 5\} \).
Now find the intersection with \( C \): \( (A \cup B) \cap C = \{1, 2, 3, 5\} \cap \{2, 3, 4, 5\} = \{2, 3, 5\} \).
\[ P(A \cup B|C) = \frac{n((A \cup B) \cap C)}{n(C)} = \frac{3}{4} \]
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Top CBSE CLASS XII Conditional Probability Questions
- If \( P(A|B) = P(A'|B) \), then which of the following statements is true?
- E and F are two independent events such that \( P(\overline{E}) = 0.6 \) and \( P(E \cup F) = 0.6 \). Find \( P(F) \) and \( P(\overline{E} \cup \overline{F}) \).
According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights.
Assume that an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are 55\%, 37\% and 17\% due to severe, moderate and light turbulence respectively.
On the basis of the above information, answer the following questions:
Find the probability that an airplane reached its destination late. If the airplane reached its destination late, find the probability that it was due to moderate turbulence.- Let \( E \) be an event of a sample space \( S \) of an experiment, then \( P(S | E) \) is:
- A pair of dice is thrown simultaneously. If \( X \) denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of \( X \).
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is