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એક સમતોલ સિક્કાને ચાર-વાર ઉછાળવામાં આવે છે અને એક વ્યક્તિ પ્રત્યેક છાપ $(H)$ પર $Rs. 1$ જીતે છે અને પ્રત્યેક કાંટા $(T) $ પ૨ $Rs.1.50$ હારે છે. આ પ્રયોગનાં નિદર્શાવકાશ પરથી શોધો કે ચાર વાર સિક્કાને ઉછાળ્યા પછી તે કેટલી ૨કમ પ્રાપ્ત કરી શકે છે તથા આ પ્રત્યેક રકમની સંભાવના શોધો.
Solution
since the coin is tossed four time, there can be a maximum of $4$ heads and tails.
When $4$ heads turns up, $\mathrm {Rs.}$ $1+$ $\mathrm {Rs.}$ $1+$ $\mathrm {Rs.}$ $1+$ $\mathrm {Rs.}$ $1=$ $\mathrm {Rs.}$ $4$ is the gain.
When $3$ heads and $1$ tail turn up, $\mathrm {Rs.}$ $1+$ $\mathrm {Rs.}$ $1+$ $\mathrm {Rs.}$ $1-$ $\mathrm {Rs.}$ $1.50=$ $\mathrm {Rs.}$ $3-$ $\mathrm {Rs.}$ $1.50= $ $\mathrm {Rs.}$ $1.50$ is the gain.
When $2$ heads and $2$ tail turn up, $\mathrm {Rs.}$ $1+$ $\mathrm {Rs.}$ $1-$ $\mathrm {Rs.}$ $1.50-$ $\mathrm {Rs.}$ $1.50=-$ $\mathrm {Rs.}$ $1,$ ie., $\mathrm {Rs.}$ $1$ is the loss.
When $1$ heads and $3$ tail turn up, $\mathrm {Rs.}$ $1-$ $\mathrm {Rs.}$ $1.50-$ $\mathrm {Rs.}$ $1.50-$ $\mathrm {Rs.}$ $1.50=-$ $\mathrm {Rs.}$ $3.50,$ i.e., $\mathrm {Rs.}$ $3.50$ is the loss.
When $4 $ tails turn up, $-$ $\mathrm {Rs.}$ $1.50-$ $\mathrm {Rs.}$ $1.50-$ $\mathrm {Rs.}$ $1.50-$ $\mathrm {Rs.}$ $1.50=-$ $\mathrm {Rs.}$ $6.00,$ ie., $\mathrm {Rs.}$ $6.00$ is the loss.
There are $2^{4}=16$ elements in the sample space $S$, which is given by:
$S =\{ HHHH ,\, HHHT ,\, HHTH$ , $HTHH ,\, THHH$, $HHTT,\, HTTH,\, TTHH$, $HTHT, \,THTH,\, THHT, \, H T T T $, $T H T T , \, T T H T ,\, T T H T $, $T T T H , \,T T T T \}$
$\therefore n( S )=16$
The person wins $\mathrm {Rs.}$ $4.00$ when $4$ heads turn up, i.e., when the event $\{HHHH\}$ occurs.
$\therefore $ Probability $($ of winning $\mathrm {Rs.}$ $4.00$ $)=\frac{1}{16}$
The person wins $\mathrm {Rs.}$ $1.50$ when $3$ heads and one tail turns up, i.e., when the event $\{HHHT, \,H H T H , \,H T H H ,\, T H H H \}$ occurs.
$\therefore $ Probability $($ of winning $\mathrm {Rs.}$ $1.50$ $)=\frac{4}{16} =\frac{1}{4}$
The person loses $\mathrm {Rs.}$ $1.00$ when $2$ heads and $2$ tails turns up, ie., when the event $\{HHTT,\, HTTH , \,T T H H $, $H T H T ,\,T H T H , \, T H H T \}$ occurs.
$\therefore $ Probability $($ of loosing $\mathrm {Rs.}$ $1.00)=\frac{6}{16}=\frac{3}{8}$
The person losses $\mathrm {Rs.}$ $3.50$ when $1$ head and $3$ tails turn up, ie., when the event $\{ HTTT,\, THTT , \,T T H T ,\,T T T H \}$ occurs.
$\therefore $ Probability $($ of loosing $\mathrm {Rs.}$ $3.50)=\frac{4}{16}=\frac{1}{4}$
The person losses $\mathrm {Rs.}$ $6.00$ when $4$ head and $3$ tails turn up, ie., when the event $\{TTTT\}$ occurs.
$\therefore $ Probability $($ of loosing $\mathrm {Rs.}$ $ 6.00)=\frac{1}{16}$