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PROBABILITY — Statistics STD 12 Commerce — Question

Gujarat BoardEnglish MediumSTD 12 CommerceStatisticsPROBABILITY4 Marks
Question
The sample data obtained about marks scored by a large group of candidates appearing for a public examination of $100$ marks are given in the following table.  $\begin{array}{|c|c|c|c|c|c|} \hline \text{Marks} & 20\ \text{or less} & 21-40 & 41-60 & 61-80 & 81-100 \\ \hline \text{No. of Candidates} & 83 & 162 & 496 & 326 & 124 \\ \hline \end{array}$
One candidate is randomly selected from those appearing for the public examination. Find the probability that this candidate has scored : $(1)$ less than $41$ marks $(2)$ More than $60$ marks $(3)$ Marks from $21$ to $80 .$
✓

Answer

The number of candidates selected in the sample is $n=83+162+496+326+124=1191$.
$(1)$ Event $A=$ The selected candidate scores less than $41$ marks.
$P(A)=$ Relative frequency for the candidates scoring less than $41$ marks.
$=\frac{\text { No. of candidates scoring less than } 41 \text { marks }}{\text { Total number of candidates in the sample }}=\frac{m}{n}$
$m=$ No. of candidates scoring less than $41$ marks
$=83+162$
$ =245$
Now, $P(A)=\frac{m}{n}$
$=\frac{245}{1191}$
Required probability $=\frac{245}{1191}$
$(2)$ Event $B=$ The selected candidate scores more than $60$ marks $P(B)=$ relative frequency for candidates scoring more than $60$ marks.
$=\frac{\text { No. of candidates scoring more than } 60 \text { marks }}{\text { Total number of candidates in the sample }}=\frac{m}{n}$
$m=$ No. of candidates scoring more than $60$ marks $=326+124$
$=450$
Now, $P(B)=\frac{m}{n}$
$ =\frac{450}{1191}$
$=\frac{150}{397}$
Required probability $=\frac{150}{397}$
$(3)$ Event $C=$ The selected candidate scores from $21$ to $80$ marks $P(C)=$ relative frequency for candidates scoring from $21$ to $80$ marks. $=\frac{m}{n}=\frac{\text { No. of candidates scoring from } 21 \text { to } 80 \text { marks }}{\text { Total number of candidates in the sample }}$
$m=$ No. of candidates scoring from $21$ to $80$ marks $=162+496+326$
$=984$
Now, $P(C)=\frac{m}{n}$
$ =\frac{984}{1191}$
$ =\frac{328}{397}$
Required probability $=\frac{328}{397}$

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