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

The probability distribution of random variable

X

is given by:

$X$	1	2	3	4	5
$P (X)$	$K$	$2 K$	$2 K$	$3 K$	$K$

Let

p = P (1 < X < 4 ∣ X < 3)

. If

5 p = λ \cdot k

then

λ

is equal to

Answer: 30

Solution

First, find

K

using the fact that probabilities sum to 1:

K + 2 K + 2 K + 3 K + K = 1 ⟹ 9 K = 1 ⟹ K = \frac{1}{9}

Now compute the conditional probability:

p = P (1 < X < 4 ∣ X < 3) = \frac{P ( 1 < X < 4 and X < 3 )}{P ( X < 3 )}

The event

1 < X < 4

and

X < 3

means

X = 2

.

P (X = 2) = 2 K = \frac{2}{9}

P (X < 3) = P (X = 1) + P (X = 2) = K + 2 K = 3 K = \frac{3}{9} = \frac{1}{3}

Therefore:

p = \frac{2/9}{1/3} = \frac{2}{9} \times 3 = \frac{2}{3}

Given

5 p = λ \cdot K

:

5 \times \frac{2}{3} = λ \times \frac{1}{9} ⟹ \frac{10}{3} = \frac{λ}{9} ⟹ λ = 30

JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Integer answer

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