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 Next

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

SA
The probability that a student buys a colouring book is 0.7 and that she buys a box of colours is 0.2. The probability that she buys a colouring book, given that she buys a box of colours, is 0.3. Find the probability that the student: (i) Buys both the colouring book and the box of colours. (ii) Buys a box of colours given that she buys the colouring book.
MCQ_SINGLE
A box contains 4 red, 5 blue and 1 green marble. A child randomly takes out a marble from the box, notes down the colour and puts it back in the box. If the activity is repeated 3 times, what is the probability that at least one marble is red?
VSA
In a village of 8000 people, 3000 go out of the village to work and 4000 are women. It is noted that 30% of women go out of the village to work. What is the probability that a randomly chosen individual is either a woman or a person working outside the village?
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.
SA
A die with number 1 to 6 is biased such that probability of $P(2)=\frac{3}{10}$ and probability of other numbers is equal. Find the mean of the number of times number 2 appears on the dice, if the dice is thrown twice.
View All Questions