There are four people in a room. For every two people, there is a 50% chance that they are friends.

Question

Contituatio of the above question......

 

Two people are connected if:

?they are friends, or
?a third person is friends with both of them, or
?they have different friends who are friends of each other.
What is the probability that every pair of people in this room is connected?

 

please answer the above question for me (ASAP)

 

Thank you

Answer 1

There are 6 possible friendships here, and then two possible connections via a third party per non-friendship. If the four people are named Ann,Bob,Carol and Dave, and AB represents that Ann and Bob are friends. Then we do not need the symbol BA, as it is redundant, since order does not matter in expressing a friendship. So the mutually exclusive friendships that have no dependency on each other are AB,AC,AD,BC,BD, and CD. If a direct friendship does not exist, between two people, they can still be connected by one and/or the other people left. So if ~AB, then A can still connect to B via either C or D or both. But again you have to be careful of repeats. If A-C-B represents that A is friends with C who is friends with B. Then A is connected to B. But ~AB is now a constraint (~AB can also be expressed as NOT AB or !AB or AB except with the bar on top). And AC and BC are also constraints. So once you make sure that A is connected with B,C and D via either direct friendship or third party friendship. You have to then consider within each set of constraints, what are the possible ways to make sure B is connected to C and D. And finally how could A,B, and C could all be connected, but still not have D connected?

 

The answer will be the sum total of the probabilities of every possible combination that results in all four people being connected via direct or third party friendship. So to start, if Ann is friends with everyone (i.e. AB, AC, and AD) then everyone is connected via Ann as a third party. This only has a 12.5% chance of happening (50% chance of AB, multiplied by the 50% chance AC will also happen, and again the 50% chance AD will concurrently happen). Then you proceed along the lines of Ann is not friends with Bob, multiplied by the probability of a third party connection.

 

In some cases, it may be easier to calculate the probability of a success by recalling that the set of all possible outcomes is the union of the set of all successful outcomes and the set of all unsuccessful outcomes. Expressing this numerically, that % chance of success, S, and % chance of failure F, summed up is 100% (all possible outcomes). So S + F =100. Or rearranged S = 100 - F. Or more simply put if you can more easily calculate the total failure rate, you can subtract it from 100 to find the total success rate.

 

I hope this gives you a good direction to start solving the problem.