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A fair coin is tossed four times, and a person win $₹ 1$ for each head and lose $₹ 1.50$ 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.
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(i) No head and 4 tail appear
Money lost $=₹ 4 \times 1.50=₹ 6.00$
There is only 1 way when, TTTT occurs
No. of exhaustive cases $=2^4=16$
$\therefore$ Probability of getting no head or 4 tails $=\frac{1}{16}$
(ii) When 1 head and 3 tails appear.
Money lost $=₹(-1 \times 1+3 \times 1.50)=₹ 3.50$
There are 4 ways when 1 head and 3 tails occur i.e.
HTTT, THTT, TTHH, TTTH
Probability of getting 1 head and 3 tails $=\frac{4}{16}=\frac{1}{4}$
(iii) When 2 head and 2 tails appear.
Money lost $=₹(2 \times 1.5-1 \times 2)$ $=₹(3-2)=₹ 1$
2 heads and 2 tails may occur as HHTT, HTHT, HTTH, THHT, THTH, TTHH Thus, 2 heads and 2 tails may appear in 6 ways $=\frac{6}{16}=\frac{3}{8}$
(iv) When 3 head 1 tail appear, Money gained $₹(3 \times 1-1 \times 1.5)=₹ 1.50$ 3 heads and 1 tail may occurs as HHHT,HHTH, HTHH, THHH
$\therefore 3$ heads and 1 tail appear in 4 ways,
$\therefore$ Probability of getting 3 heads and 1 tail $=\frac{4}{16}=\frac{1}{4}$
(v) When all the heads appear, money gained $=₹ 4 \times 1=₹ 4$
4 Head occur as $H H H H$ i.e., in one way;
$\therefore$ Probability of getting 4 heads $=\frac{1}{16}$
Money lost $=₹ 4 \times 1.50=₹ 6.00$
There is only 1 way when, TTTT occurs
No. of exhaustive cases $=2^4=16$
$\therefore$ Probability of getting no head or 4 tails $=\frac{1}{16}$
(ii) When 1 head and 3 tails appear.
Money lost $=₹(-1 \times 1+3 \times 1.50)=₹ 3.50$
There are 4 ways when 1 head and 3 tails occur i.e.
HTTT, THTT, TTHH, TTTH
Probability of getting 1 head and 3 tails $=\frac{4}{16}=\frac{1}{4}$
(iii) When 2 head and 2 tails appear.
Money lost $=₹(2 \times 1.5-1 \times 2)$ $=₹(3-2)=₹ 1$
2 heads and 2 tails may occur as HHTT, HTHT, HTTH, THHT, THTH, TTHH Thus, 2 heads and 2 tails may appear in 6 ways $=\frac{6}{16}=\frac{3}{8}$
(iv) When 3 head 1 tail appear, Money gained $₹(3 \times 1-1 \times 1.5)=₹ 1.50$ 3 heads and 1 tail may occurs as HHHT,HHTH, HTHH, THHH
$\therefore 3$ heads and 1 tail appear in 4 ways,
$\therefore$ Probability of getting 3 heads and 1 tail $=\frac{4}{16}=\frac{1}{4}$
(v) When all the heads appear, money gained $=₹ 4 \times 1=₹ 4$
4 Head occur as $H H H H$ i.e., in one way;
$\therefore$ Probability of getting 4 heads $=\frac{1}{16}$
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