Probability of Random Coincidence

The keep coming back response from people whom read the article is that the agreement between the lattice positions and quantum numbers is a coincidence. This might be true but the odds are extremely low. Following a conservative approach the probability of a random coincidence is:

P(random coincidence) = 6.03 10^-16

How this value was calculated?

The probability of choosing a random number from a group of numbers which have the same probability is

where n is the number of the numbers in the group.

Assuming that the group of the numbers contains integers up to the number of the different states (predicted by quantum theory) then the probability of choosing the correct number for the allowed states can be calculated in the same way.

This approach is relatively conservative since does not take into account that the quantum numbers might be chosen from a group of numbers containing higher integers then the number of possible states and that the allowed states have to be equally distributed among the possible states. The number of different quantum states in a lattice model is represented by the number of the different lattice positions. Since all the quantum numbers has to be satisfied by the lattice model the overall probability of the random coincidence between the number of lattice positions and the number of quantum states can be calculated by multiplying the probabilities of the individual trials. Assuming that the overall number of the quantum states contains four main, seven angular momentum and seven magnetic momentum numbers, the probability of the random coincidence of these numbers is

Substituting the numbers of possible states allow calculating the probability of the random coincidence.

This extremely small probability almost excludes the possibility of random coincidence. This probability also allows calculating the probability of the correctness of the model, which is practically one.

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