JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Probability

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A student appeared in an examination consisting of 8 true - false type questions. The student guesses the answers with equal probability. The smallest value of n, so that the probability of guessing at least '

n

' correct answers is less than

\frac{1}{2}

, is :

(A)5✓ Correct
(B)6
(C)3
(D)4

Answer: (A)

Solution

The number of correct guesses

X

is binomial with

8

trials and probability

\frac{1}{2}

, so

P (X \geq n) = \frac{1}{2 ^{8}} k = n \sum 8 (k 8)

For

n = 4

P (X \geq 4) = \frac{70 + 56 + 28 + 8 + 1}{256} = \frac{163}{256} > \frac{1}{2}

For

n = 5

P (X \geq 5) = \frac{56 + 28 + 8 + 1}{256} = \frac{93}{256} < \frac{1}{2}

Since

P (X \geq n)

decreases as

n

increases, every smaller

n

also fails, so the smallest value of

n

5

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Single correct option

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