The random variable X has the following probability distribution where a and b are some constants : X 1 2 3 4 5 --- --- --- --- --- --- P(X) 0.2 a a 0.2 b If the mean E(X)=3, then find values of a and b and hence determine P(X≥3).

MathsCBSE Class 12 · Board exam 2024, set 65/5/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2024: question 31

Set in the CBSE Class 12 board exam of 2024, paper 65/5/1, section C, question 31, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31 · 3 marks · Short answerProbability

The random variable X has the following probability distribution where a and b are some constants :

X	1	2	3	4	5
P(X)	0.2	a	a	0.2	b

If the mean $E (X) = 3$ , then find values of a and b and hence determine $P (X \geq 3)$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/5/1.

Hints

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

Hint 1

Total probability is 1 — this gives one equation.

Hint 2

The mean $\sum x_{i} P (x_{i}) = 3$ gives the second equation.

Hint 3

$P (X \geq 3) = 1 - P (X = 1) - P (X = 2)$ .

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.

$E (X) = \sum x P (x) = 0.2 + 2 a + 3 a + 0.8 + 5 b = 5 a + 5 b + 1$ .
½ mark
$\sum p_{i} = 1$ : $0.2 + a + a + 0.2 + b = 1 \Rightarrow 2 a + b = 0.6$ .
½ mark
$E (X) = 3 \Rightarrow 5 a + 5 b = 2 \Rightarrow a + b = 0.4$ . Subtracting from $2 a + b = 0.6$ : $a = 0.2 = \frac{1}{5}$ , $b = 0.2 = \frac{1}{5}$ .
1 mark
$P (X \geq 3) = 1 - [P (X = 1) + P (X = 2)] = 1 - (0.2 + 0.2) = 0.6 = \frac{3}{5}$ .
1 mark

Answer: $a = \frac{1}{5}$ , $b = \frac{1}{5}$ , $P (X \geq 3) = \frac{3}{5}$

Where the marks are

Expression for $E (X)$ (½ mark)
Sum of probabilities gives $2 a + b = 0.6$ (½ mark)
Solves to $a = b = \frac{1}{5}$ (1 mark)
$P (X \geq 3) = \frac{3}{5}$ (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 paper2024 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 31

Hints

Step-by-step solution

Revise the chapter

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