Class CBSE Class 12 Mathematics Probability Q #911
COMPETENCY BASED
APPLY
3 Marks 2023 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 $X$.
Prev Next

AI Tutor Explanation

Powered by Gemini

Step-by-Step Solution

Let $S$ be the sample space when a pair of dice is thrown. Then $n(S) = 6 \times 6 = 36$.

Let $X$ be the random variable denoting the absolute difference of numbers obtained on the pair of dice. The possible values of $X$ are $0, 1, 2, 3, 4, 5$.

We need to find the probability distribution of $X$, i.e., $P(X=x)$ for $x = 0, 1, 2, 3, 4, 5$.

  • $P(X=0)$: The pairs with absolute difference 0 are (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). So, there are 6 such pairs. $P(X=0) = \frac{6}{36} = \frac{1}{6}$
  • $P(X=1)$: The pairs with absolute difference 1 are (1,2), (2,1), (2,3), (3,2), (3,4), (4,3), (4,5), (5,4), (5,6), (6,5). So, there are 10 such pairs. $P(X=1) = \frac{10}{36} = \frac{5}{18}$
  • $P(X=2)$: The pairs with absolute difference 2 are (1,3), (3,1), (2,4), (4,2), (3,5), (5,3), (4,6), (6,4). So, there are 8 such pairs. $P(X=2) = \frac{8}{36} = \frac{2}{9}$
  • $P(X=3)$: The pairs with absolute difference 3 are (1,4), (4,1), (2,5), (5,2), (3,6), (6,3). So, there are 6 such pairs. $P(X=3) = \frac{6}{36} = \frac{1}{6}$
  • $P(X=4)$: The pairs with absolute difference 4 are (1,5), (5,1), (2,6), (6,2). So, there are 4 such pairs. $P(X=4) = \frac{4}{36} = \frac{1}{9}$
  • $P(X=5)$: The pairs with absolute difference 5 are (1,6), (6,1). So, there are 2 such pairs. $P(X=5) = \frac{2}{36} = \frac{1}{18}$

The probability distribution of $X$ is:

$X$ 0 1 2 3 4 5
$P(X)$ $\frac{1}{6}$ $\frac{5}{18}$ $\frac{2}{9}$ $\frac{1}{6}$ $\frac{1}{9}$ $\frac{1}{18}$

Correct Answer: See explanation for probability distribution table.<\/strong>

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply their understanding of probability and absolute difference to construct the probability distribution.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a series of steps to calculate the probabilities for each possible value of the random variable X. This involves a specific algorithm or method.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question requires the student to apply the concepts of probability to a specific scenario, demonstrating their ability to use the knowledge in a practical context rather than just recalling definitions or formulas.

More from this Chapter

MCQ_SINGLE
18. The probability that A speaks the truth is $\frac{4}{5}$ and that of B speaking the truth is $\frac{3}{4}$. The probability that they contradict each other in stating the same fact 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.
MCQ_SINGLE
If A and B are events such that \(P(A/B)=P(B/A)\ne0,\) then :
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.
SA
Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.
View All Questions