List of top Mathematics Questions on Probability and Statistics asked in UP Board XII

There are 10 black and 5 white balls in a bag. Two balls are taken out, one after another, and the first ball is not placed back before the second is taken out. Assume that the drawing of each ball from the bag is equally likely. What is the probability that both the balls drawn are black? 

Find the probability of 53 Sundays in a leap year.
If \( y = Ae^x + B \) where \( A, B \) are constants, then show that \( \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0 \).

The probabilities of solving a question by \( A \) and \( B \) independently are \( \frac{1}{2} \) and \( \frac{1}{3} \) respectively. If both of them try to solve it independently, find the probability that:

 

 

If \( A \) and \( B \) are independent events, where \( P(A) = \frac{3}{10} \), \( P(B) = \frac{6}{10} \), then find \( P(A \cup B) \) and \( P(A \cap B) \).

The given events \( A \) and \( B \) are such that \( P(A) = \frac{1}{4} \), \( P(B) = \frac{1}{2} \), and \( P(A \cap B) = \frac{1}{8} \); then find \( P(A' \cap B') \).

(d) If \( P(A) = 0.5 \), \( P(B) = 0.4 \), and \( A \) and \( B \) are independent events, then find the values of (A) \( P(A \cap B) \) and (B) \( P(A \cup B) \).
(b) If \( \sin^{-1} \left( \frac{1}{2} \right) = \tan^{-1} x \), then find the value of \( x \).

(d) If \( P(A) = \frac{3}{13} \), \( P(B) = \frac{5}{13} \), and \( P(A \cap B) = \frac{2}{13} \), find the value of \( P(B | A) \):

A die is thrown two times. It is found that the sum of the appeared numbers is 6. Find the conditional that the number 4 appeared at least one time.

There are two children in a family. If it is known that at least one child is a boy, find the  that both children are boys.

(b) Two coins are tossed together. Find the probability of getting both tails:

A die is thrown twice. Let us represent the event 'obtaining an odd number on the first throw' by A and the event 'obtaining an odd number on the second throw' by B. Test the independency of the events A and B.

If the position vectors of the vertices \( A, B, C \) of a triangle \( \Delta ABC \) are \( \vec{a}, \vec{b}, \vec{c} \), respectively, then prove that the area of \( \Delta ABC \) is \[ \frac{1}{2} \left| \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} \right|. \]
(a) If a die is thrown three times, find the probability of getting one appearing number in them will be odd:
A die is thrown once. The number on the die is a multiple of 3 is denoted by \( E \), and the number on the die is even is denoted by \( F \). Are \( E \) and \( F \) independent events?
The probability of \( A \) winning the race is \( \frac{1}{3} \) and that of \( B \) is \( \frac{1}{4} \). In this race, find the probability that neither \( A \) nor \( B \) can win the race.

If \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A | B) = \frac{3}{10} \), find \( P(A \cup B) \).

A die is thrown once. If E represents the event ‘the number obtained on the die is a multiple of 3’ and F represents the event ‘the number obtained on the die is even’, then tell whether the events E and F are independent.

A die was thrown twice and it was found that the sum of the numbers that appeared was 6. Find the conditional probability that the number 4 appeared at least once.