JEE MAIN - Mathematics (2020 - 7th January Morning Slot - No. 18)
An unbiased coin is tossed 5 times. Suppose that a variable X is assigned the value of k when k
consecutive heads are obtained for k = 3, 4, 5, otherwise X takes the value -1. Then the expected
value of X, is :
$$ - {3 \over {16}}$$
$$ - {1 \over 8}$$
$${1 \over 8}$$
$${3 \over {16}}$$
Explanation
Number of ways 3 consecutive heads can appers
(1) HHHT_
(2) _THHH
(3) THHHT
$$ \therefore $$ Probablity of getting 3 consecutive heads
= $${2 \over {32}}$$ + $${2 \over {32}}$$ + $${1 \over {32}}$$ = $${5 \over {32}}$$
Number of ways 4 consecutive heads can appers
(1) HHHHT
(2) THHHH
$$ \therefore $$ Probablity of getting 4 consecutive heads
= $${1 \over {32}}$$ + $${1 \over {32}}$$ = $${2 \over {32}}$$
Number of ways 5 consecutive heads can appers
(1) HHHHH
$$ \therefore $$ Probablity of getting 5 consecutive heads
= $${1 \over {32}}$$
Now Probablity of getting 0, 1, and 2 consecutive heads
= 1 - $$\left( {{5 \over {32}} + {2 \over {32}} + {1 \over {32}}} \right)$$ = $${{{24} \over {32}}}$$
Now, Expectation
= (-1) $$ \times $$ $${{{24} \over {32}}}$$ + 3 $$ \times $$ $${{{5} \over {32}}}$$ + 4 $$ \times $$ $${{{2} \over {32}}}$$ + 5 $$ \times $$ $${{{1} \over {32}}}$$
= $${1 \over 8}$$
(1) HHHT_
(2) _THHH
(3) THHHT
$$ \therefore $$ Probablity of getting 3 consecutive heads
= $${2 \over {32}}$$ + $${2 \over {32}}$$ + $${1 \over {32}}$$ = $${5 \over {32}}$$
Number of ways 4 consecutive heads can appers
(1) HHHHT
(2) THHHH
$$ \therefore $$ Probablity of getting 4 consecutive heads
= $${1 \over {32}}$$ + $${1 \over {32}}$$ = $${2 \over {32}}$$
Number of ways 5 consecutive heads can appers
(1) HHHHH
$$ \therefore $$ Probablity of getting 5 consecutive heads
= $${1 \over {32}}$$
Now Probablity of getting 0, 1, and 2 consecutive heads
= 1 - $$\left( {{5 \over {32}} + {2 \over {32}} + {1 \over {32}}} \right)$$ = $${{{24} \over {32}}}$$
Now, Expectation
= (-1) $$ \times $$ $${{{24} \over {32}}}$$ + 3 $$ \times $$ $${{{5} \over {32}}}$$ + 4 $$ \times $$ $${{{2} \over {32}}}$$ + 5 $$ \times $$ $${{{1} \over {32}}}$$
= $${1 \over 8}$$
Comments (0)
