(1) Let pi be the probability that a randomly chosen point has i many friends, i = 0, 1, 2, 3, 4.

Question

Consider the 6 × 6 square in the figure. Let A₁, A₂,…., A₄₉ 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.

(1) 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 _____.

(2) Two distinct points are chosen randomly out of the points A₁, A₂, ……., A₄₉. Let p be the probability that they are friends. Then the value of 7p is _____.

Answer 1

(1) 

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 × 4 = 20

Number of points having 4 friends = 49 – 24 = 25

(2) Number of ways of selecting 2 adjacent dots in I row = 6

Similarly, number of ways of selecting 2 adjacent dots in I column = 6

\(\therefore\) Number of ways of selecting 2 adjacent dots from the matrix = 6 × 7 + 6 × 7 = 84