# Can you have imaginary probability?

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So say I have a bag of 10 marbles, 1 red, blue, green, yellow, purple, orange, magenta, cyan, brown and gray, and there's a 1/i change of me picking a green one at any given second. What is the likelihood I will pick a green number as a percentage of 10? How much time could pass before I would actually be able to pick a green one?

Actually, let's just say I have a bag of 10 marbles, they're all green and the probability of picking one is 1/i. What is the percentage I'd pick a green one using that probability?

Edited by questionposter

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Probabilities are real numbers, AFAIK.

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Probabilities are real numbers, AFAIK.

I know, but what does it mean if I say I have a 1/i chance of picking a green marble?

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I know, but what does it mean if I say I have a 1/i chance of picking a green marble?

It means nothing if probabilities are real numbers. It's akin to saying the mass of an elephant is blue.

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It means nothing if probabilities are real numbers. It's akin to saying the mass of an elephant is blue.

So if it means nothing, would I never pick up a green marble if the chance of picking one up was 1/i?

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So if it means nothing, would I never pick up a green marble if the chance of picking one up was 1/i?

Saying the chance is 1/i has no meaning.

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I don't see any trouble with generalising to complex numbers, in the sense of complex valued "probability" densities, after all this is just integration of complex functions; so we could employ lots of good notions from complex analysis. The problem is, as swantsont points out, I have no idea how to interpret a "complex probability".

That said, I would not be surprised is someone has written about using "complex probabilities" in stochastic processes or quantum mechanics. There they may have some interpretation, but it won't be as simple probabilities. If I get time I will see what I can find.

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So say I have a bag of 10 marbles, 1 red, blue, green, yellow, purple, orange, magenta, cyan, brown and gray, and there's a 1/i change of me picking a green one at any given second. What is the likelihood I will pick a green number as a percentage of 10? How much time could pass before I would actually be able to pick a green one?

Actually, let's just say I have a bag of 10 marbles, they're all green and the probability of picking one is 1/i. What is the percentage I'd pick a green one using that probability?

You already said so. Percent means /100; therefore 1/i is 100/100i or 100/i %, for each second.

And to the other poster swansont, who wasn't so good, yes there can be sheerly imaginary probability. Probability is the amplitude conjugate product: When amplitude is imaginary, such as elèctric currend for elèctric charge, the probability is negative. If you work backwards and set the probability to be imaginary (i), the amplitude is 22/2 + 22i/2. The imaginary merely represents another linearly independent factor.

Edited by alysdexia

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Actually, let's just say I have a bag of 10 marbles, they're all green and the probability of picking one is 1/i. What is the percentage I'd pick a green one using that probability?

Then I'd say your ability to calculate the probability is flawed. In a bag full of green marbles the probability of picking a green marble is a probability of 1.

OTOH, the probability amplitude in quantum mechanics is a complex number.

Edited by doG

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Then I'd say your ability to calculate the probability is flawed. In a bag full of green marbles the probability of picking a green marble is a probability of 1.

OTOH, the probability amplitude in quantum mechanics is a complex number.

But this is a special bag.

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Then I'd say your ability to calculate the probability is flawed. In a bag full of green marbles the probability of picking a green marble is a probability of 1.

OTOH, the probability amplitude in quantum mechanics is a complex number.

But probability amplitude is not probability. That comes from the modulus squared, as stated in the link.

You already said so. Percent means /100; therefore 1/i is 100/100i or 100/i %, for each second.

And to the other poster swansont, who wasn't so good, yes there can be sheerly imaginary probability. Probability is the amplitude conjugate product: When amplitude is imaginary, such as elèctric currend for elèctric charge, the probability is negative. If you work backwards and set the probability to be imaginary (i), the amplitude is 22/2 + 22i/2. The imaginary merely represents another linearly independent factor.

Yes, it's the product; if you have an imaginary amplitude, you get a real number when you square it. I have no idea what your example purports to show, I don't know what 22/2 + 22i/2 means. Can you explain it?

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But probability amplitude is not probability. That comes from the modulus squared, as stated in the link.

Yes, it's the product; if you have an imaginary amplitude, you get a real number when you square it. I have no idea what your example purports to show, I don't know what 22/2 + 22i/2 means. Can you explain it?

show, -> show;

That's the square root of 2, duh, over 2 plus the square root of 2 times i over 2. If you know the Pýthagorean theorem and vector analýsis, you'd see that as the componends of a unit imaginary vector's square root. Roots of the imaginary trend towards the unity.

Edited by alysdexia

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$\sqrt{2}/2 + i\sqrt{2}/2$ gives a real result when multiplied by its complex conjugate. So I still don't know what you're talking about.

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If you pick someone randomly from the UK's population there's a roughly 1 in 70 million chance it's me.

If you pick 2 people then there's a roughly 2 in 70 million chance and so on.

For small numbers of people, the probability is pretty close to proportional to the number of people you pick.

If you pick i people is the probability that it's me i/70,000,000 ?

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$\sqrt{2}/2 + i\sqrt{2}/2$ gives a real result when multiplied by its complex conjugate. So I still don't know what you're talking about.

It's a square root, not a conjugate root.

$i = (a+bi)(a-bi)$

$= (a^2+b^2)$

$= (a+\sqrt{i-a^2}i)(a-\sqrt{i-a^2}i)$

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If you pick someone randomly from the UK's population there's a roughly 1 in 70 million chance it's me.

If you pick 2 people then there's a roughly 2 in 70 million chance and so on.

For small numbers of people, the probability is pretty close to proportional to the number of people you pick.

If you pick i people is the probability that it's me i/70,000,000 ?

I'm just not sure how you could pick imaginary discrete objects like people. Conceptually, it's very hard to picture.

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I'm just not sure how you could pick imaginary discrete objects like people. Conceptually, it's very hard to picture.

True, but that was my point. The maths may make sense but the physical system it describes is meaningless.

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I'm just not sure how you could pick imaginary discrete objects like people. Conceptually, it's very hard to picture.

It's not hard nor soft. If you pick i people, you pick something other than people.

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It's not hard nor soft. If you pick i people, you pick something other than people.

Which leads me to ask, if I'm interested in 1 of Britain's 70 million people, why do I pick something other than people?

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Hi,

Have a look at this paper

Quantum Mechanics as Complex Probability Theory

S. Youssef

(Submitted on 2 Jul 1993 (v1), last revised 14 Feb 1994 (this version, v2))

Realistic quantum mechanics based on complex probability theory is shown to have a frequency interpretation, to coexist with Bell's theorem, to be linear, to include wavefunctions which are expansions in eigenfunctions of Hermitian operators and to describe both pure and mixed systems. Illustrative examples are given. The quantum version of Bayesian inference is discussed

Regards,

Wil

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Hi,

Have a look at this paper

Quantum Mechanics as Complex Probability Theory

S. Youssef

(Submitted on 2 Jul 1993 (v1), last revised 14 Feb 1994 (this version, v2))

Realistic quantum mechanics based on complex probability theory is shown to have a frequency interpretation, to coexist with Bell's theorem, to be linear, to include wavefunctions which are expansions in eigenfunctions of Hermitian operators and to describe both pure and mixed systems. Illustrative examples are given. The quantum version of Bayesian inference is discussed

Regards,

Wil

so...particles have properties of existence that involves imaginary number, but we can't see those properties directly because they are imaginary numbers and we can't see their value...but we can see when...something like probability multiplies by another imaginary number to produce real numbers and thus "real" effects? I guess this makes the term "real" a lot more subjective.

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so...particles have properties of existence that involves imaginary number, but we can't see those properties directly because they are imaginary numbers and we can't see their value...but we can see when...something like probability multiplies by another imaginary number to produce real numbers and thus "real" effects? I guess this makes the term "real" a lot more subjective.

One way that complex quantities can enter is through the use of the Fourier Transform, the momentum and position space representations have a transform pair

relationship. But when you do the final calculation for the probability in either case, you use the complex conjugates so that the probabilities are always real.

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One way that complex quantities can enter is through the use of the Fourier Transform, the momentum and position space representations have a transform pair

relationship. But when you do the final calculation for the probability in either case, you use the complex conjugates so that the probabilities are always real.

Well of course because the only way they could possibly measure probability is if all the numbers are real, you can't see "i" apples or draw a marble with a 1/i chance, but I think what should be explained more is the mechanism for which those imaginary numbers occur in nature in the first place, probably related to polar coordinates or waves derived from a unit circle on a complex plane, but how does slapping a circle on a piece of paper create particles?

Edited by EquisDeXD

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