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## Summary:

- The probabilities of mutually exclusive events are additive. For bosons this does not seem to be the case. How can we explain this?

## Main Question or Discussion Point

This questions was brought to my attention by Kazu Okayasu.

According to probability theory the probabilities of mutually exclusive events add upp.

As an example we can distribute 2 balls in two boxes with two compartments each.

So there's a box on the left with a lower and an upper compartment and there's an identical box on the right.

The probability that both balls are in the box on the left is 1/4.

If we now also investigate in which compartment each ball is, we find that there are 4 different ways to place the 2 balls in the two compartments of the box on the left, each with a probability of 1/16.

This is in agreement with probability theory: All distributions are mutually exclusive and indeed the 4 probabilities 1/16 add up to 1/4.

For bosons the situation is different since they are indistinguishable.

The probability of finding 2 bosons in the box on the left is 1/3, but there are 3 different ways to place two bosons in the two compartments of the box on the left, each with a probability of 1/10.

These probabilities add up to 3/10 and not 1/3.

How can we understand this?

Does it mean that different distributions of bosons in compartments are not mutually exclusive?

That would mean that there is some sort of overlap between the event of 2 bosons being in the upper compartment and the event of 1 being in the upper and 1 in the lower, somehow both distributions can exist at the same time.

According to probability theory the probabilities of mutually exclusive events add upp.

As an example we can distribute 2 balls in two boxes with two compartments each.

So there's a box on the left with a lower and an upper compartment and there's an identical box on the right.

The probability that both balls are in the box on the left is 1/4.

If we now also investigate in which compartment each ball is, we find that there are 4 different ways to place the 2 balls in the two compartments of the box on the left, each with a probability of 1/16.

This is in agreement with probability theory: All distributions are mutually exclusive and indeed the 4 probabilities 1/16 add up to 1/4.

For bosons the situation is different since they are indistinguishable.

The probability of finding 2 bosons in the box on the left is 1/3, but there are 3 different ways to place two bosons in the two compartments of the box on the left, each with a probability of 1/10.

These probabilities add up to 3/10 and not 1/3.

How can we understand this?

Does it mean that different distributions of bosons in compartments are not mutually exclusive?

That would mean that there is some sort of overlap between the event of 2 bosons being in the upper compartment and the event of 1 being in the upper and 1 in the lower, somehow both distributions can exist at the same time.

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