Question

A fair coin is tossed four times, and a person win $Rs1$ for each head and lose $Rs1.50$ for each tail that turns up. From the sample space calculate how many different amounts of money you can have after four tosses and the probability of having each of these amounts.

Answer 1

A fair coin is tossed four times, then the sample space is
$S=\{HHHH,HHHT,HHTH,HHTT,THHH,HTHH,HTHT,HTTH,HTTT,THHT,THTH,THTT,TTHH,TTHT,TTTH,TTTT\}$

Results	Profit on Heads	Loss on tails	Amount of money
$4\text{head}0\text{tail}$	$4\times 1=4$	$0\times 1.5=0$	$4-0=4$
$3\text{head}1\text{tail}$	$3\times 1=3$	$1\times 1.5=1.5$	$3-1.5=1.5$
$2\text{head}2\text{tail}$	$2\times 1=2$	$2\times 1.5=3$	$2-3=-1$
$1\text{head}3\text{tail}$	$1\times 1=1$	$3\times 1.5=4.5$	$1-4.5=-3.5$
$0\text{head}4\text{tail}$	$0\times 1=0$	$4\times 1.5=6$	$0-6=-6$

Hence, the different amount of money is $\{4,1.5,-1,-3.5,-6\}$

Number of $4\text{heads and}0\text{tail is}n\left(A\right)=1$
Number of $3\text{heads and}1\text{tail is}n\left(B\right)=4$
Number of $2\text{heads and}2\text{tail is}n\left(C\right)=6$
Number of $1\text{heads and}3\text{tail is}n\left(D\right)=4$
Number of $0\text{heads and}4\text{tail is}n\left(E\right)=1$
$n\left(S\right)=16$

$P\text{(winning of 4 rupee)}=\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$

$P\text{(winning of 4 rupee)}=\frac{n\left(A\right)}{n\left(S\right)}=\frac{1}{16}$

$P\text{(winning of 1.5 rupee)}=\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$

$P\text{(winning of 1.5 rupee)}=\frac{n\left(B\right)}{n\left(S\right)}=\frac{4}{16}=\frac{1}{4}$

$P\text{(loss of 1 rupee)}=\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$

$P\text{(loss of 1 rupee)}=\frac{n\left(C\right)}{n\left(S\right)}=\frac{6}{16}=\frac{3}{8}$

$P\text{(loss of 3.5 rupee)}=\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$

$P\text{(loss of 3.5 rupee)}=\frac{n\left(D\right)}{n\left(S\right)}=\frac{4}{16}=\frac{1}{4}$

$P\text{(loss of 6 rupee)}=\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$

$P\text{(loss of 6 rupee)}=\frac{n\left(E\right)}{n\left(S\right)}=\frac{1}{16}$

Hence the required probability that winning of $4$ rupee is $\frac{1}{16}$,

Hence the required probability that winning of $1.5$ rupee is $\frac{1}{4}$,

Hence the required probability that loss of $1$ rupee is $\frac{3}{8}$,

Hence the required probability that loss of $3.5$ rupee is $\frac{1}{4}$,

Hence the required probability that loss of $6$ rupee is $\frac{1}{16}$.