Testbook Mock Test (DM + Probability)

Question

Ram draws an empty graph of 6 nodes and tosses a fair die for every pair of distinct nodes, if a prime number occurs he draws an edge between the respective pair otherwise he moves on without drawing edge between that pair. What is the probability that after the end of the experiment a randomly chosen node will have degree 2?

(a) 5/16

(b) 1/2

(c) 3/8

(d) None of the above

Comments

@Rahul,

but dont you think we should multiply with 6 in the end as it can be any vertex and this is probability of one vertex

multiplying by 6 effect is cancelled, see how.

select one vertex in $\binom{6}{1}$ , and probabilty of selecting that vertex is $\frac{1}{6}$,

so overall it is $\binom{6}{1}$*$\frac{1}{6}$*(rest of the expression)

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@joshi_nitesh: What are we choosing here in 5C2?

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Every node have 5 adjacent nodes, so taking any two among them hence making the degree = 2

Answer 1

This problem reduce to find the triangle with 6 vertices. there must be the three node with either 3 or 5 outcomes (3,3,5) or (5,5,3) or  more but not more than that, if the prime outcomes is more than three it will lead to K4, K5, or K6 so we only focus on K3 which is a cycle of three edges.

we know that the number of cycle 3 with n vertices is n(n-1)(n-2)/6 = 6*5*4/5 = 20
and total number of graph formed with probability is (1+1)^6 = 64

so 20/64 = 5/16