If the sample space of an experiment is S-1- -2-3- then which of the following assignment of probabilities is valid-

The correct option is D P(ω_1)=12, P(ω_2)=13, P(ω_3)=16 S=(ω _ 1 ,ω _ _ 2 ,ω _ 3 ) (i) probability of any event cannot be negative (ii) su ...

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Standard X

Mathematics

Addition Theorem of Probability for 2 Events

If the sample...

Question

If the sample space of an experiment is $S = (ω_{1}, ω_{2}, ω_{3})$ , then which of the following assignment of probabilities is valid?

P (ω_{1}) = \frac{1}{2}, P (ω_{2}) = \frac{1}{3}, P (ω_{3}) = \frac{2}{3}

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P (ω_{1}) = \frac{1}{2}, P (ω_{2}) = \frac{1}{3}, P (ω_{3}) = \frac{1}{4}

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P (ω_{1}) = \frac{1}{2}, P (ω_{2}) = \frac{1}{3}, P (ω_{3}) = \frac{- 1}{6}

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P (ω_{1}) = \frac{1}{2}, P (ω_{2}) = \frac{1}{3}, P (ω_{3}) = \frac{1}{6}

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Solution

The correct option is D $P (ω_{1}) = \frac{1}{2}, P (ω_{2}) = \frac{1}{3}, P (ω_{3}) = \frac{1}{6}$
$S = (ω_{1}, ω_{_{2}}, ω_{3})$
$(i)$ probability of any event cannot be negative
$(i i)$ sum of probability of all the events in sample space is equal to $1$
By the above axioms,
Option A, B,C are false since it violates either two mentioned axioms.
Option D: $P (ω_{1}) = \frac{1}{2}, P (ω_{2}) = \frac{1}{3}, P (ω_{2}) = \frac{1}{6} P (ω_{1}), P (ω_{2}), P (ω_{3}) > 0 \to A x i o m (i) s a t i s f i e d$
$P (ω_{1}), P (ω_{2}) + P (o m e g a_{3}) = \frac{1}{3} + \frac{1}{3} + \frac{1}{6} = 1 \to A x i o m (i i) a l s o s a t i s f i e d$
$∴$ correct option is D

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Similar questions

Q. Which of the following can not be valid assignment of probabilities for outcomes of sample space S = {

{ω_{1}, ω_{2}, ω_{3}, ω_{4}, ω_{5}, ω_{6}, ω_{7}}

Assignment	$ω_{1}$	$ω_{2}$	$ω_{3}$	$ω_{4}$	$ω_{5}$	$ω_{6}$	$ω_{7}$
(a)	0.1	0.01	0.05	0.03	0.01	0.2	0.6
(b)	$\frac{1}{7}$	$\frac{1}{7}$	$\frac{1}{7}$	$\frac{1}{7}$	$\frac{1}{7}$	$\frac{1}{7}$	$\frac{1}{7}$
(c)	0.1	0.2	0.3	0.4	0.5	0.6	0.7
(d)	-0.1	0.2	0.3	0.4	-0.2	0.1	0.3
(e)	$\frac{1}{14}$	$\frac{2}{14}$	$\frac{3}{14}$	$\frac{4}{14}$	$\frac{5}{14}$	$\frac{6}{14}$	$\frac{15}{14}$

Which of the following can not be valid assignment of probabilities for outcomes of sample space S =

Assignment	ω₁	ω₂	ω₃	ω₄	ω₅	ω₆	ω₇
(a)	0.1	0.01	0.05	0.03	0.01	0.2	0.6
(b)
(c)	0.1	0.2	0.3	0.4	0.5	0.6	0.7
(d)	–0.1	0.2	0.3	0.4	–0.2	0.1	0.3
(e)

Q. Following are the margins of victory in the football matches of a league.

1, 3, 2, 5, 1, 4, 6, 2, 5, 2, 2, 2, 4, 1, 2, 3, 1, 1, 2, 3, 2, 6, 4, 2, 1, 1, 4, 2, 1, 5, 3, 3, 2, 3, 2, 4, 2, 1, 2, 3

, find the mode of this data.

Question 02

Which of the following options does not represent ground state electronic configuration of an atom ?

(a) $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{8} 4 s^{2}$ (b) $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{9} 4 s^{2}$ (c) $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{10} 4 s^{1}$ (d) $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{5} 4 s^{1}$

Q. The value of the expression

(1 + \frac{1}{ω}) (1 + \frac{1}{ω^{2}}) + (2 + \frac{1}{ω}) (2 + \frac{1}{ω^{2}}) + (3 + \frac{1}{ω^{2}}) (3 + \frac{1}{ω^{2}}) + . . . . . . . . . . . . . + (n + \frac{1}{ω}) (n + \frac{1}{ω^{2}})

(

ω

is the root of unity) is

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If the sample space of an experiment is S=(ω1,ω2,ω3), then which of the following assignment of probabilities is valid?

If the sample space of an experiment is $S = (ω_{1}, ω_{2}, ω_{3})$ , then which of the following assignment of probabilities is valid?