Let a sample space be S- 1- 2- 6-Which of the following assignments of probabilities to each outcome are valid-Outcomes- 1- 2- 3- 4- 5- 6a161616161616b100000c18231313-14-13d11211216161632e0-10-20-30-40-50-6

Name: NCERT - Grade 11 - Mathematics - Probability - Q32
Uploaded: 2022-07-04T10:36:42+05:30
Description: Check Assignment of probability Condition (i) : Each of the number p(ω_𝑖) is positive and less than one. Condition (ii) nbsp; : Sum of probabilities of each ...

Check Assignment of probability Condition (i) : Each of the number p(ω_𝑖) is positive and less than one. Condition (ii) nbsp; : Sum of probabilities of each ...

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Byju's Answer

Standard XII

Mathematics

Fundamental Principle of Counting

Let a sample ...

Question

Let a sample space be $S = {ω_{1}, ω_{2}, . . ., ω_{6}}$ .Which of the following assignments of probabilities to each outcome are valid $?$

Outcomes $ω_{1}$ $ω_{2}$ $ω_{3}$ $ω_{4}$ $ω_{5}$ $ω_{6}$

$(a)$ $\frac{1}{6}$ $\frac{1}{6}$ $\frac{1}{6}$ $\frac{1}{6}$ $\frac{1}{6}$ $\frac{1}{6}$

$(b)$ $1$ $0$ $0$ $0$ $0$ $0$

$(c)$ $\frac{1}{8}$ $\frac{2}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $- \frac{1}{4}$ $- \frac{1}{3}$

$(d)$ $\frac{1}{12}$ $\frac{1}{12}$ $\frac{1}{6}$ $\frac{1}{6}$ $\frac{1}{6}$ $\frac{3}{2}$

$(e)$ $0.1$ $0.2$ $0.3$ $0.4$ $0.5$ $0.6$

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Solution

Check Assignment of probability

Condition $(i)$ : Each of the number $p (ω_{i})$ is positive and less than one.

Condition $(i i)$ : Sum of probabilities of each outcome of an event $= 1$

$(a)$ Each of the number $p (ω_{i})$ is positive and less than one.
Sum of probabilities $(S O P) = ω_{1} + ω_{2} + ω_{3} + ω_{4} + ω_{5} + ω_{6}$
$= \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{6}{6} = 1$
Therefore, Assignment $(a)$ is valid.

$(b)$ Each of the number $p (ω_{i})$ is either $0 or 1$ .
Sum of probabilities $= (1 + 0 + 0 + 0 + 0 + 0 + 0 = 1)$
Therefore, Assignment $(b)$ is valid.

$(c)$ Probability of any event cannot be negative.
$($ probability of events $ω_{5} and ω_{6}$ are negetive $)$
Therefore, assignment $(c)$ is not valid.

$(d)$ Probability of any event cannot be greater than $1$ .
Since, $p (ω_{6}) = \frac{3}{2} > 1$
Therefore, assignment $(d)$ is not valid.

$(e)$ Sum of probabilities $(S O P)$
$= ω_{1} + ω_{2} + ω_{3} + ω_{4} + ω_{5} + ω_{6}$
$= 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 = 2.1$
Since, sum of probabilities of events is greater than $1$ .
So, assignment $(e)$ is not valid.

Suggest Corrections

Outcomes	$ω_{1}$	$ω_{2}$	$ω_{3}$	$ω_{4}$	$ω_{5}$	$ω_{6}$
$(a)$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$
$(b)$	$1$	$0$	$0$	$0$	$0$	$0$
$(c)$	$\frac{1}{8}$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$- \frac{1}{4}$	$- \frac{1}{3}$
$(d)$	$\frac{1}{12}$	$\frac{1}{12}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{3}{2}$
$(e)$	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$

Let a sample space be S={ω1,ω2,...,ω6}.Which of the following assignments of probabilities to each outcome are valid? Outcomes ω1 ω2 ω3 ω4 ω5 ω6 (a) 16 16 16 16 16 16 (b) 1 0 0 0 0 0 (c) 18 23 13 13 −14 −13 (d) 112 112 16 16 16 32 (e) 0.1 0.2 0.3 0.4 0.5 0.6