The random variable X has a probability distribution P(X) of the following form, where 'k' is some real number: P(X)=casesk, & if x=0\\ 2k, & if x=1\\ 3k, & if x=2\\ 0, & otherwisecases (i) Determine the value of k. (ii) Find P(X 2).

MathsCBSE Class 12 · Sample paper, session 2023-24 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2023-24: question 27

From the official CBSE sample question paper for 2023-24, section C, question 27, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27 · 3 marks · Short answerProbability

The random variable $X$ has a probability distribution $P (X)$ of the following form, where 'k' is some real number:
$P (X) = ⎩ ⎨ ⎧ k, 2 k, 3 k, 0, if x = 0 if x = 1 if x = 2 otherwise$

(i)
Determine the value of $k$ .
0 marks
(ii)
Find $P (X < 2)$ .
0 marks
(iii)
Find $P (X > 2)$ .
0 marks

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

The total of all the probabilities in a distribution is 1.

Hint 2

$X < 2$ means $X = 0$ or $X = 1$ . Which values of $X$ are greater than 2?

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 add to 1: $k + 2 k + 3 k = 1 \Rightarrow 6 k = 1 \Rightarrow k = \frac{1}{6}$ .
1 mark
(ii) $P (X < 2) = P (X = 0) + P (X = 1) = k + 2 k = 3 k = 3 \times \frac{1}{6} = \frac{1}{2}$ .
1 mark
(iii) $X$ takes no value greater than 2, so $P (X > 2) = 0$ .
1 mark

Answer: (i) $k = \frac{1}{6}$ ; (ii) $\frac{1}{2}$ ; (iii) $0$

Where the marks are

Uses $\sum P (X) = 1$ to get $k = \frac{1}{6}$ (1 mark)
$P (X < 2) = 3 k = \frac{1}{2}$ (1 mark)
$P (X > 2) = 0$ (1 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 paperSample paper 2023-24: every question, by section.

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CBSE Class 12 Mathematics sample paper 2023-24: question 27

Hints

Step-by-step solution

Revise the chapter

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