Class CBSE Class 12 Mathematics Probability Q #1412
KNOWLEDGE BASED
REMEMBER
3 Marks 2025 AISSCE(Board Exam) SA
A coin is tossed twice. Let X be a random variable defined as number of heads minus number of tails. Obtain the probability distribution of X and also find its mean.
Prev Next

AI Tutor Explanation

Powered by Gemini

Detailed Solution

Step 1: Define the Sample Space

When a coin is tossed twice, the sample space S is given by: S = {HH, HT, TH, TT}

Step 2: Define the Random Variable X

Let X be the random variable representing the number of heads minus the number of tails.

Step 3: Determine the Values of X for each outcome

For each outcome in the sample space, we calculate the value of X: - HH: 2 heads, 0 tails. X = 2 - 0 = 2 - HT: 1 head, 1 tail. X = 1 - 1 = 0 - TH: 1 head, 1 tail. X = 1 - 1 = 0 - TT: 0 heads, 2 tails. X = 0 - 2 = -2

Step 4: Determine the Probability Distribution of X

The possible values of X are -2, 0, and 2. We calculate the probabilities for each value: - P(X = -2) = P(TT) = 1/4 - P(X = 0) = P(HT) + P(TH) = 1/4 + 1/4 = 2/4 = 1/2 - P(X = 2) = P(HH) = 1/4 The probability distribution of X is: X | -2 | 0 | 2 ------- | -------- | -------- | -------- P(X) | 1/4 | 1/2 | 1/4

Step 5: Calculate the Mean of X

The mean (or expected value) of X, denoted by E(X), is calculated as: E(X) = Σ [x * P(x)] E(X) = (-2 * 1/4) + (0 * 1/2) + (2 * 1/4) E(X) = -2/4 + 0 + 2/4 E(X) = 0

Final Answer: The probability distribution is: X: -2, 0, 2 with probabilities 1/4, 1/2, 1/4 respectively. The mean is 0.

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because it requires recalling the definition of probability distribution and the formula for calculating the mean of a random variable.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concepts of random variables, probability distributions, and expected value (mean). It involves applying these concepts to a specific scenario (coin toss).
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly relates to the syllabus content on probability distributions and random variables. The question tests the student's understanding of basic definitions and formulas.

More from this Chapter

SA
The probability distribution for the number of students being absent in a class on a Saturday is as follows: X: 0, 2, 4, 5; $P(X)$: p, 2p, 3p, p. Where X is the number of students absent. (i) Calculate p. (ii) Calculate the mean of the number of absent students on Saturday.
SA
Two dice are thrown. Defined are the following two events A and B: $A=\{(x,y):x+y=9\}$, $B=\{(x,y):x\ne3\}$ where (x, y) denote a point in the sample space. Check if events A and B are independent or mutually exclusive.
MCQ_SINGLE
A coin is tossed and a card is selected at random from a well shuffled pack of 52 playing cards. The probability of getting head on the coin and a face card from the pack 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.
LA
(a) In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/5 be the probability that he knows the answer and 2/5 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/3. What is the probability that the student knows the answer, given that he answered it correctly? OR (b) A box contains 10 tickets, 2 of which carry a prize of ₹8 each, 5 of which carry a prize of ₹4 each, and remaining 3 carry a prize of ₹2 each. If one ticket is drawn at random, find the mean value of the prize.
View All Questions