The probability distribution of a random variable X is given below : X 1 2 3 --- --- --- --- P(X) k/2 k/3 k/6 (i) Find the value of k. (ii) Find P(1 ≤ X < 3). (iii) Find E(X), the mean of X.

(i) k=1; (ii) 56; (iii) E(X)=53

MathsCBSE Class 12 · Board exam 2023, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2023: question 30(a)

Set in the CBSE Class 12 board exam of 2023, paper 65/1/1, section C, question 30(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30(a) · 3 marks · Short answerProbability

The probability distribution of a random variable X is given below :

X	1	2	3
P(X)	$\frac{k}{2}$	$\frac{k}{3}$	$\frac{k}{6}$

(i)
Find the value of k.
0 marks
(ii)
Find $P (1 \leq X < 3)$ .
0 marks
(iii)
Find $E (X)$ , the mean of X.
0 marks

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

The total probability of a distribution is 1.

Hint 2

$1 \leq X < 3$ means $X = 1$ or $X = 2$ . The mean is $\sum x_{i} P (x_{i})$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

(i) The probabilities must add to 1: $\frac{k}{2} + \frac{k}{3} + \frac{k}{6} = 1$ .
1 mark
(i) This simplifies to $k \cdot \frac{3 + 2 + 1}{6} = k = 1$ .
½ mark
(ii) $P (1 \leq X < 3) = P (X = 1) + P (X = 2) = \frac{k}{2} + \frac{k}{3} = \frac{5 k}{6} = \frac{5}{6}$ .
½ mark
(iii) $E (X) = \sum p_{i} x_{i} = 1 \cdot \frac{k}{2} + 2 \cdot \frac{k}{3} + 3 \cdot \frac{k}{6} = \frac{k}{2} + \frac{2 k}{3} + \frac{k}{2} = \frac{5 k}{3}$ .
½ mark
(iii) With $k = 1$ : $E (X) = \frac{5}{3}$ .
½ mark

Answer: (i) $k = 1$ ; (ii) $\frac{5}{6}$ ; (iii) $E (X) = \frac{5}{3}$

Where the marks are

Sets the sum of probabilities equal to 1 (1 mark)
Gets $k = 1$ (½ mark)
$P (1 \leq X < 3) = \frac{5}{6}$ (½ mark)
Sets up $\sum p_{i} x_{i} = \frac{5 k}{3}$ (½ mark)
$E (X) = \frac{5}{3}$ (½ mark)

Revise the chapter

Chapter: Probability (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Probability, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2023: question 30(a)

Hints

Step-by-step solution

Revise the chapter

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