JEE MAIN - Mathematics (2019 - 12th January Evening Slot - No. 2)
In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :
$${{400} \over 3}$$ loss
0
$${{400} \over 9}$$ loss
$${{400} \over 3}$$ gain
Explanation
Expected Gain/Loss =
= w $$ \times $$ 100 + Lw($$-$$ 50 + 100)
+ L2w ($$-$$ 50 $$-$$ 50 + 100) + L3($$-$$ 150)
= $${1 \over 3} \times 100 + {2 \over 3}.{1 \over 3}\left( {50} \right) + {\left( {{2 \over 3}} \right)^2}\left( {{1 \over 3}} \right)\left( 0 \right)$$
$$ + {\left( {{2 \over 3}} \right)^3}\left( { - 150} \right) = 0$$
here w denotes probability that outcome 5 or 6 (w = $${2 \over 6} = {1 \over 3}$$)
here L denotes probability that outcome
1,2,3,4 (L = $${4 \over 6}$$ = $${2 \over 3}$$)
= w $$ \times $$ 100 + Lw($$-$$ 50 + 100)
+ L2w ($$-$$ 50 $$-$$ 50 + 100) + L3($$-$$ 150)
= $${1 \over 3} \times 100 + {2 \over 3}.{1 \over 3}\left( {50} \right) + {\left( {{2 \over 3}} \right)^2}\left( {{1 \over 3}} \right)\left( 0 \right)$$
$$ + {\left( {{2 \over 3}} \right)^3}\left( { - 150} \right) = 0$$
here w denotes probability that outcome 5 or 6 (w = $${2 \over 6} = {1 \over 3}$$)
here L denotes probability that outcome
1,2,3,4 (L = $${4 \over 6}$$ = $${2 \over 3}$$)
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