Question

each of the $5{ }$ sides and the $5{ }$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color can be expressed as , where are relatively prime positive integers. what is ?

Answer 1

The required probability for  triangle whose vertices are among the vertices of the pentagon is 253/256 = 0.988

What is probability ?

Probability is the ratio of number of favorable outcomes to the total number of outcomes .It can not be more than 1.

According to given data in the question:

There are 3 cases happen ,they are 4 all are of same color,  3 are one color and one is the other, 2 are one color and 2 are of the other color.

following all cases,

Number of favorable outcomes = 12

Total number of outcomes = 2^10 = 1024

probability that there will be a triangle whose vertices are not among the vertices of the pentagon.

probability = 12/1024  

required probability = 1 - 12/1024 = 253/256

Hence, the required answer is 253/256.

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