JEE Main · 2020 · 07 Jan 2020 · Shift 1 · Probability

An unbiased coin is tossed 5 times. Suppose that a variable

X

is assigned the value

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

Answer: (C)

Solution

Let

k

be the longest run of heads in five tosses;

X = k

when

k \geq 3

and

X = - 1

otherwise, over

32

equally likely sequences. Counting runs gives

P (X = 3) = \frac{5}{32}

P (X = 4) = \frac{2}{32}

P (X = 5) = \frac{1}{32}

, and the remaining

\frac{24}{32}

carry

X = - 1

. Hence

E (X) = \frac{- 24 + 3 ( 5 ) + 4 ( 2 ) + 5 ( 1 )}{32} = \frac{4}{32} = \frac{1}{8}

JEE Main · 2020 · 07 Jan 2020 · Shift 1 · Single correct option