Class CBSE Class 12 Mathematics Probability Q #1805
COMPETENCY BASED
APPLY
3 Marks 2026 AISSCE(Board Exam) SA
A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected. Find P(A and B) and find whether the events A and B are independent events or not.
Prev Finish

AI Tutor Explanation

Powered by Gemini

Detailed Solution

Step 1: Define the Sample Space

Since two cards are drawn with replacement from 6 cards (numbered 1 to 6), the total number of outcomes is $6 \times 6 = 36$.

Step 2: Analyze Event A

Event A is the sum of numbers being 10. The possible pairs are (4, 6), (5, 5), and (6, 4). Thus, $P(A) = \frac{3}{36} = \frac{1}{12}$.

Step 3: Analyze Event B

Event B is getting a number other than 4 on the first card. The first card can be 1, 2, 3, 5, or 6 (5 options). The second card can be any of the 6 numbers. Thus, $n(B) = 5 \times 6 = 30$. $P(B) = \frac{30}{36} = \frac{5}{6}$.

Step 4: Calculate P(A and B)

Event (A and B) means the sum is 10 AND the first card is not 4. From the pairs in A {(4, 6), (5, 5), (6, 4)}, only (5, 5) and (6, 4) satisfy the condition of the first card not being 4. Thus, $P(A \cap B) = \frac{2}{36} = \frac{1}{18}$.

Step 5: Check for Independence

Events are independent if $P(A \cap B) = P(A) \times P(B)$. Here, $P(A) \times P(B) = \frac{1}{12} \times \frac{5}{6} = \frac{5}{72}$. Since $\frac{1}{18} \neq \frac{5}{72}$, the events are not independent.

Final Answer: P(A and B) = 1/18; Events are not independent.

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must translate the verbal description of events into set-theoretic notation and apply the definition of independent events.
Knowledge Dimension: PROCEDURAL
Justification: The problem requires a step-by-step algorithmic approach to calculate probabilities and verify a mathematical condition.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the core understanding of Probability (Chapter 13) beyond rote memorization, requiring logical deduction of sample space subsets.

More from this Chapter

MCQ_SINGLE
A box has 4 green, 8 blue and 3 red pens. A student picks up a pen at random, checks its colour and replaces it in the box. He repeats this process 3 times. The probability that at least one pen picked was red is:
SA
For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data it was revealed that two third of the total applicants were females and other were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.
SA
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.
SA
A pair of dice is thrown simultaneously. If $X$ denotes the absolute difference of numbers obtained on the pair of dice, then find the probability distribution of $X$.
SA
A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that, Robotic Knee replacement surgeries have 90% success rate. On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that: (i) exactly one surgery is successful, (ii) at most two surgeries are successful.
View All Questions