SATHEE: Probability 3 Question 45

Probability 3 Question 45

45. Of the three independent events $E_{1}, E_{2}$ and $E_{3}$ , the probability that only $E_{1}$ occurs is $α$ , only $E_{2}$ occurs is $β$ and only $E_{3}$ occurs is $γ$ . Let the probability $p$ that none of events $E_{1}, E_{2}$ or $E_{3}$ occurs satisfy the equations

$(α - 2 β), p = α β$ and $(β - 3 γ) p = 2 β γ$ . All the given probabilities are assumed to lie in the interval $(0, 1)$ .

Then, $\frac{probability of occurrence of E_{1}}{probability of occurrence of E_{3}}$ is equal to

Passage Type Questions

Passage

Football teams $T_{1}$ and $T_{2}$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_{1}$ winning, drawing and losing a game against $T_{2}$ are $\frac{1}{2}, \frac{1}{6}$ and $\frac{1}{3}$ , respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T_{1}$ and $T_{2}$ , respectively, after two games.

(2016 Adv.)

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Answer:

Correct Answer: 45. (c)

Solution:

PLAN

Forthe events to be independent,

$\begin{aligned} P (E_{1} \cap E_{2} \cap E_{3}) & = P (E_{1}) \cdot P (E_{2}) \cdot P (E_{3}) \\ P (E_{1} \cap {\bar{E}}_{2} \cap {\bar{E}}_{3}) & = P (only E_{1} occurs) \\ = P (E_{1}) \cdot (1 - P (E_{2})) (1 - P (E_{3})) \end{aligned}$

Let $x, y$ and $z$ be probabilities of $E_{1}, E_{2}$ and $E_{3}$ , respectively.

$\begin{array}{lll} ∴ & α = x (1 - y) (1 - z) \\ β = (1 - x) \cdot y (1 - z) \\ \Rightarrow & γ = (1 - x) (1 - y) z \\ p = (1 - x) (1 - y) (1 - z) \end{array}$

Given, $(α - 2 β) p = α β$ and $(β - 3 γ) p = 2 β γ$

From above equations, $x = 2 y$ and $y = 3 z$

$\begin{array}{ll} ∴ & x = 6 z \\ \Rightarrow & \frac{x}{z} = 6 \end{array}$

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Probability 3 Question 45

45. Of the three independent events $E_{1}, E_{2}$ and $E_{3}$ , the probability that only $E_{1}$ occurs is $α$ , only $E_{2}$ occurs is $β$ and only $E_{3}$ occurs is $γ$ . Let the probability $p$ that none of events $E_{1}, E_{2}$ or $E_{3}$ occurs satisfy the equations

Passage Type Questions

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Probability 3 Question 45

45. Of the three independent events E1,E2 and E3, the probability that only E1 occurs is α, only E2 occurs is β and only E3 occurs is γ. Let the probability p that none of events E1,E2 or E3 occurs satisfy the equations

Passage Type Questions

Passage

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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45. Of the three independent events $E_{1}, E_{2}$ and $E_{3}$ , the probability that only $E_{1}$ occurs is $α$ , only $E_{2}$ occurs is $β$ and only $E_{3}$ occurs is $γ$ . Let the probability $p$ that none of events $E_{1}, E_{2}$ or $E_{3}$ occurs satisfy the equations