Consider the 6 × 6 square in the figure. Let A 1 , A 2 , … , 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.…

Question

Consider the 6 &#215; 6 square in the figure. Let A 1 , A 2 , &#8230; , 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. (There are two questions based on PARAGRAPH &#160; &#34; II &#34; , the question given below is one of them) 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 &#160; E ( X ) is

Aicharya · Accepted Answer

Plotting the diagram of the given data we get,
Now from above diagram we get,
Number of points having $0$ friend $= 0$
Number of points having $1$ friend $= 0$
Number of points having $2$ friends $= 4$
Number of points having $3$ friends $= 5 \times 4 = 20$
Number of points having $4$ friends $= 49 - 24 = 25$
So, probability distribution table will be,
Now from above table expected value will be given by,
$7 \cdot E (X) = 7 (0 + 0 + \frac{4}{49} \times 2 + \frac{20}{49} \times 3 + \frac{25}{49} \times 4)$
$\Rightarrow 7 \cdot E (X) = 7 (\frac{100 + 60 + 8}{49})$
$\Rightarrow 7 \cdot E (X) = 24$

Consider the 6 × 6 square in the figure. Let A 1 , A 2 , … , A 49 be the points of intersections…

Solution

Consider the 6 &#215; 6 square in the figure. Let A 1 , A 2 , &#8230; , A 49 be the points of intersections…

Solution

Consider the 6 × 6 square in the figure. Let A 1 , A 2 , … , A 49 be the points of intersections…