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$S=HHHH,HHHT,HHTH,HTHH,THHH,HHTT,HTTH,TTHH,HTHT,THTH,THHT,HTTT,THTT,TTHT,TTTH,TTTT$

$β΄n(S)=16$

Since, the coin is tossed four times there can be a maximum of $4$ heads .

When $4$ heads turns up, $Re1+Re1+Re1+Re1=Rs4$ is the gain.

When $3$ heads and $1$ tail turn up, $Re1+Re1+Re1βRs1.50=Rs3βRs1.50=Rs1.50$ is the gain.

When $2$ heads and $2$ tails turns up, $Re1+Re1βRs1.50βRs1.50=βRe1$, i.e., Re $1$ Β is the loss.

When $1$ head and $3$ tails turn up,$Re1βRs1.50βRs1.50βRs.1.50=βRs3.50$, i.e.,Β Rs $3.50$ is the loss.

When $4$ tails turn up, $βRs1.50βRs1.50βRs1.50βRs1.50=βRs6.00$ i.e., Rs $6.00$ is the loss.

The person wins Rs $4.00$ when $4$ heads turn up i.e., when the event ${HHHH}$ occurs.

$β΄$Probability (of winning Rs $4.00$)=Β $161β$

When $4$ heads turns up, $Re1+Re1+Re1+Re1=Rs4$ is the gain.

When $3$ heads and $1$ tail turn up, $Re1+Re1+Re1βRs1.50=Rs3βRs1.50=Rs1.50$ is the gain.

When $2$ heads and $2$ tails turns up, $Re1+Re1βRs1.50βRs1.50=βRe1$, i.e., Re $1$ Β is the loss.

When $1$ head and $3$ tails turn up,$Re1βRs1.50βRs1.50βRs.1.50=βRs3.50$, i.e.,Β Rs $3.50$ is the loss.

When $4$ tails turn up, $βRs1.50βRs1.50βRs1.50βRs1.50=βRs6.00$ i.e., Rs $6.00$ is the loss.

The person wins Rs $4.00$ when $4$ heads turn up i.e., when the event ${HHHH}$ occurs.

$β΄$Probability (of winning Rs $4.00$)=Β $161β$

The person wins Rs $1.50$ when $3$ heads and one tail turn up i.e., when the event ${HHHT,HHTH,HTHH,THHH}$ occurs

$β΄$Probability (of winning Rs $1.50$) =Β $164β=41β$

The person loses Re $1.00$ when $2$ heads and $2$ tails turn up i.e., when the event ${HHTT,HTTH,TTHH,HTHT,THTH,THHT}$ occurs

$β΄$ Probability (of losing Re 1.00) =Β $166β=83β$

The person loses Rs $3.50$ when $1$ head and $3$ tails turn up i.e., when the event ${HTTT,THTT,TTHT,TTTH}$ occurs

Probability (of losing Rs $3.50$) =Β $164β=41β$

The person loses Rs $6.00$ when $4$ tails turn up i.e., when the event ${TTTT}$ occurs

Probability (of losing Rs $6.00$) =Β $161β$Β

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