JEE MAIN - Mathematics (2017 - 8th April Morning Slot - No. 12)
An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :
$${{255} \over {256}}$$
$${{127} \over {128}}$$
$${{63} \over {64}}$$
$${{1} \over {2}}$$
Explanation
An unbiased coin is tossed 8 times which is same as 8 coins tossed 1 times.
$$\therefore\,\,\,$$ Possible no. of out come = 28
$$\therefore\,\,\,$$ Sample space = 28
Here in this condition, all head or all tail out come is not acceptable.
No. of times all head can occur
(H H H H H H H H) = 1
$$\therefore\,\,\,$$ Probability (all head) = $${1 \over {{2^8}}}$$ = $${1 \over {256}}$$
No. of times all tail can occur
(T T T T T T T T) = 1
$$\therefore\,\,\,$$ Probability (all tail) = $${1 \over {{2^8}}}$$ = $${{1 \over {256}}}$$
$$\therefore\,\,\,$$ Required probability
= 1 $$-$$ (P (All head) + P (All tail))
= 1 $$-$$ ( $${{1 \over {256}}}$$ + $${{1 \over {256}}}$$)
= 1 $$-$$ $${{1 \over {128}}}$$
= $${{127} \over {128}}$$
$$\therefore\,\,\,$$ Possible no. of out come = 28
$$\therefore\,\,\,$$ Sample space = 28
Here in this condition, all head or all tail out come is not acceptable.
No. of times all head can occur
(H H H H H H H H) = 1
$$\therefore\,\,\,$$ Probability (all head) = $${1 \over {{2^8}}}$$ = $${1 \over {256}}$$
No. of times all tail can occur
(T T T T T T T T) = 1
$$\therefore\,\,\,$$ Probability (all tail) = $${1 \over {{2^8}}}$$ = $${{1 \over {256}}}$$
$$\therefore\,\,\,$$ Required probability
= 1 $$-$$ (P (All head) + P (All tail))
= 1 $$-$$ ( $${{1 \over {256}}}$$ + $${{1 \over {256}}}$$)
= 1 $$-$$ $${{1 \over {128}}}$$
= $${{127} \over {128}}$$
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