JEE Advanced 2023 · Paper 2 · Mathematics Q16

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MathematicsNumerical+3Question 16 of 51

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PARAGRAPH "II"

Consider the $6\times6$ square in the figure. Let $A_1, A_2, \ldots, A_{49}$ 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.

A 6 x 6 square grid of unit cells: 7 horizontal and 7 vertical lines, with a dot at each of the 49 intersection points.

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 $7E(X)$ is _____.

Your answer

Answer and worked solution

Official answer: 23.99

In the $7\times7$ array of points, count each point's friends:

Points	Number	Friends
corners	4	2
other boundary points	$4\times5=20$	3
interior points	$5\times5=25$	4

$E(X)=\frac{4\cdot2+20\cdot3+25\cdot4}{49}=\frac{168}{49}=\frac{24}{7},\qquad7E(X)=24.$

Topics:Random variables and distributions Fundamental principle of counting

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JEE Advanced 2023 · Paper 2, Mathematics question 16