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I've confused myself, conditional probability question

Anonymous Fri 08 Nov 23:11:13 2019 No.11128956 View Reply Original Report

Quoted By: >>11128988 >>11129028

Flip coin A and coin B. Heads are 1, tails are 0.
If I look at A, that has no bearing on B. B has a 50% chance of being heads or tails.

However,
P(A+B=2 | A+B>=1) = 1/3
P(B | A+B>=1) = 2/3
implies that if I observe coin A to be heads (satisfying the condition), I can conclude the other coin has a 2/3 chance of being tails. I know this isn't correct, but I can't figure out what part of the math is incorrect.

What am I missing?

Anonymous

Anonymous Fri 08 Nov 2019 23:14:44 No.11128962 Report

Quoted By: >>11128978

Each coin flip has a 50% chance of being heads or tails.

One single coin flip will have 100% either heads or tails though.

Many coin flips in aggregate will resemble 50% though.

B has a 50% chance of either way but A and B together have a higher chance of both.

Because:

A = heads or tails
B = heads or tails

So, it's more rare that both A and B would both get heads, but both A and B themselves each have 50% odds of either way.

Anonymous

Anonymous Fri 08 Nov 2019 23:20:33 No.11128978 Report

Quoted By:

>>11128962
To add on to this.

You either have

Heads heads
Heads Tails
Tails Heads
or
Tails Tails.

But if you keep going

Heads Heads Heads 3/0
Heads Heads Tails 2/1
Heads Tails Tails 1/2
Heads Tails Heads 2/1
Tails Tails Tails 0/3
Tails Tails Heads 1/2
Tails Heads Heads 2/1
Tails Heads Tails 1/2

There are less options where it is just 100% one way or the other.

So the more trials the closer you are to 50%.

Anonymous

Anonymous Fri 08 Nov 2019 23:23:42 No.11128988 Report

Quoted By:

>>11128956
Lmao you Monty Hall'ed yourself.
A=1 isn't the same as A+B>=1

Anonymous

Anonymous Fri 08 Nov 2019 23:38:24 No.11129020 Report

Quoted By:

11128956
P(B | A+B>=1) = 2/3, but
P(B | A+B>=1 and A=1) = 1/2. If you have seen coin A land heads, your state of knowledge is the latter, not the former.

Conditional probabilities conditional on A+B>=1 apply only when ALL you know about A and B is that A+B>=1. And that's quite an uncommon situation to find yourself in. Usually you know more than that.

Anonymous

Anonymous Fri 08 Nov 2019 23:44:52 No.11129028 Report

Quoted By:

>>11128956
they aren't dependent on each other is what you're missing

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