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

SA
A die is rolled. Consider events: $A=\{1,2,5\}$, $B=\{3,5\}$, $C=\{2,3,4,5\}$ and hence find: (i) $P(A|C)$ and $P(C|A)$ (ii) $P(A\cap B|C)$ and $P(A\cup B|C)$.
SA
The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that (i) target is hit (ii) atleast one shot misses the target.
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
The chances of P, Q and R getting selected as CEO of a company are in the ratio 4: 1: 2 respectively. The probabilities for the company to increase its profits from the previous year under the new CEO, P, Q or R are 0-3, 0-8 and 0.5 respectively. If the company increased the profits from the previous year, find the probability that it is due to the appointment of R as CEO.
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.
View All Questions