List of top Mathematics Questions on Conditional Probability asked in CBSE CLASS XII

An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The caps to be distributed are in three separate boxes, Box I has 30 red caps, Box II has 20 red and 10 green caps, and Box III has 30 green caps. The probability that a Box i is selected and a cap picked out is i/6, where i = 1, 2, 3.  
Based on the above information, answer the following questions :

Find the vector and cartesian equations of the line passing through the point of intersection of the lines \( \vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j}) \) and \( \vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + 3\hat{k}) \) and parallel to the line \( \frac{x - 1}{-2} = \frac{7 - y}{-3} = z \).
In a school, the probability of holding a debate competition is $\frac{1}{3}$ and that of a quiz competition is $\frac{2}{3}$. In the two participating teams, A has 4 girls and 6 boys and B has 7 girls and 3 boys. If a debate competition is held, the students are selected from team A and for the quiz competition they are selected from team B. If only two students are to be chosen from the teams, then find the probability that one will be a girl and the other a boy.
A die is rolled. Consider the following events:
$A = \{1, 2, 5\}$, $B = \{3, 5\}$, $C = \{2, 3, 4, 5\}$
Find the following conditional probabilities:
(i) $P(A|C)$ and $P(C|A)$
(ii) $P(A \cap B|C)$ and $P(A \cup B|C)$}
Mother, Father and Son line up at random for a family picture. Let events E : Son on one end and F : Father in the middle. Find \(P(E/F)\).
For two events $A$ and $B$ such that $P(A) \neq 0$ and $P(B) \neq 1$, the conditional probability $P(A'/B')$ is equal to:

A gardener wanted to plant vegetables in his garden. Hence he bought 10 seeds of brinjal plant, 12 seeds of cabbage plant, and 8 seeds of radish plant. The shopkeeper assured him of germination probabilities of brinjal, cabbage, and radish to be 25%, 35%, and 40% respectively. But before he could plant the seeds, they got mixed up in the bag and he had to sow them randomly.

What is the probability that it is a cabbage seed, given that the chosen seed germinates?
If \( P(A|B) = P(A'|B) \), then which of the following statements is true?