E and F are two independent events such that P(E) = 0.6 and P(E F) = 0.6. Find P(F) and P(E F).

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

CBSE Class 12 Mathematics 2024: question 31

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, 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

E and F are two independent events such that $P (\overline{E}) = 0.6$ and $P (E \cup F) = 0.6$ . Find $P (F)$ and $P (\overline{E} \cup \overline{F})$ .

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

Hints

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

Hint 1

First find P(E) from $P (\overline{E})$ .

Hint 2

For independent events, $P (E \cap F) = P (E) P (F)$ ; use it in the addition theorem.

Hint 3

$\overline{E} \cup \overline{F}$ is the complement of $E \cap F$ .

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.

$P (\overline{E}) = 0.6 \Rightarrow P (E) = 1 - 0.6 = 0.4$
½ mark
$P (E \cup F) = P (E) + P (F) - P (E \cap F)$ , and $P (E \cap F) = P (E) P (F)$ [E, F independent]
½ mark
$0.6 = 0.4 + P (F) - 0.4 P (F) \Rightarrow 0.6 P (F) = 0.2 \Rightarrow P (F) = \frac{1}{3}$
1 mark
$P (\overline{E} \cup \overline{F}) = P (\overline{E \cap F}) = 1 - P (E \cap F)$ [De Morgan's law]
½ mark
$= 1 - 0.4 \times \frac{1}{3} = 1 - \frac{2}{15} = \frac{13}{15}$
½ mark

Answer: P(F) = 1/3; $P (\overline{E} \cup \overline{F}) = 13/15$

Where the marks are

Finds P(E) = 0.4 (0.5 mark)
Uses addition law with independence (0.5 mark)
Solves P(F) = 1/3 (1 mark)
Uses De Morgan: $1 - P (E \cap F)$ (0.5 mark)
Gets 13/15 (0.5 mark)

Revise the chapter

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

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 65/1/1: 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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