एक अनभिनत सिक्के को चार बार उछाला जाता है और एक व्यक्ति प्रत्येक चित्त पर एक रू जीतता है और प्रत्येक पट् पर $1.50$ रू हारता है। इस परीक्षण के प्रतिदर्श समष्टि से ज्ञात कीजिए कि आप चार उछालों में कितनी विभिन्न राशियाँ प्राप्त कर सकते हैं। साथ ही इन राशियों में से प्रत्येक की प्रायिकता भी ज्ञात कीजिए ?

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}$