JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Probability XII

Passage

Consider the

6 \times 6

square in the figure. Let

A_{1}, A_{2}, \dots, A_{49}

be the points of intersections (dots in the picture) in some order. We say that

A_{i}

and

A_{j}

are friends if they are adjacent along a row or along a column. Assume that each point

A_{i}

has an equal chance of being chosen.

Consider the

6 \times 6

square in the figure. Let

A_{1}, A_{2}, \dots, A_{49}

be the points of intersections (dots in the picture) in some order. We say that

A_{i}

and

A_{j}

are friends if they are adjacent along a row or along a column. Assume that each point

A_{i}

has an equal chance of being chosen.

Let

p_{i}

be the probability that a randomly chosen point has

i

many friends,

i = 0, 1, 2, 3, 4

. Let

X

be a random variable such that for

i = 0, 1, 2, 3, 4

, the probability

P (X = i) = p_{i}

. Then the value of

7 E (X)

Answer: 24

Solution

The

6 \times 6

array of segments has

7 \times 7 = 49

lattice points. A point's number of friends is its number of grid neighbours: the

4

corners have

2

each, the

20

non-corner edge points have

3

each, and the

25

interior points have

4

each. Hence

E (X) = \frac{4 \cdot 2 + 20 \cdot 3 + 25 \cdot 4}{49} = \frac{168}{49} = \frac{24}{7}

, so

7 E (X) = 24

JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Numerical answer