Question

A fair coin is tossed four times, and a person win Re 1 for each head and lose Rs 1.50 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

Here a coin is tossed four times. The sample space S is
$S=\{HHHH,HHHT,HHTH,HTHH,HTTH,HTHT,HHTT,HTTT,THHH,THHT,THTH,TTHH,TTTH,TTHT,THTT,TTTT\}$
Amounts:
(i) for 4 heads
= 1 + 1 + 1 + 1
= Rs. 4 = winning Rs. 4
(ii) For 3 heads 1 tail
= 1 + 1 + 1 -1.50
= Rs. 1.50 = winning Rs. 1.50
(iii) For 2 heads 2 tails
= 1 + 1 - 1.50 - 1.50
= Rs. (-1) = losing Rs. 1
(iv) For 1 head 3 tails
= 1 - 1.50 - 1.50 - 1.50
= Rs. (-3.50) = losing Rs. 3.50
(v) For 4 tails = -1.50 - 1.50 - 1.50 - 1.50
= losing Rs. 6
Thus the sample space of amounts is
$=\{4,1.50,1.50,1.50,1.50,-1,-1,-1,-1,-1,-1,-3.50,-3.50,-3.50,-6\}$
Now P (winning Rs. 4) $\frac{1}{16}$
P (winning Rs. 1.50) $=\frac{4}{16}=\frac{1}{4}$
P(Losing Rs. 1) $=\frac{6}{16}=\frac{3}{8}$
P (Losing Rs. 3.50) $=\frac{4}{16}=\frac{1}{4}$
P (Losing Rs. 6) $=\frac{1}{16}$