Simple probability discrepancy

Question

This to me seems like a really stupid doubt I have but here goes anyway.

Problem:Find the probability of choosing 1 AND 2 from the numbers 1,2

Here, we take 2 possible routes-choosing 1 then 2 OR 2 then 1.

For either route the equation becomes (1/2)*(1/1)=(1/2) However, taking into account both routes, we double to get (1/2)*2=1 which is trivially true.

Take another problem:Find the probability of choosing 2 AND 2 from the numbers 2,2.

Following a similar method to the first problem, we take 2 routes-choosing 2 then 2 OR 2 then 2(because even though they have same magnitude they are still different numbers)

For either route the equation becomes (2/2)*(1/1)=(1)

However, taking into account both routes, we double to get 1*2=2 which is trivially wrong.

Why is this?

Answer 1

You confused the notions and ended up with a conundrum.

Here is a restatement of your second problem to make it clear. I have 2 balls in a bag, both marked as $2$, but one is green and one is yellow. Find the probability of choosing these 2 balls if we draw 2 balls without replacement.

There are 2 paths to drawing the 2 balls: green + yellow or yellow + green, with the probability of taking each path individually at $1/2$ you end up with $$\frac12 + \frac12 = 1$$ as in your previous case.

Here, you differentiate between the two kinds of 2-labels correctly...

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An elaboration of your mistake. If you take $2$ and $2$ in the bag to be same magnitude but different numbers (i.e. one is yellow and one is green, as in my example above), your probability of getting the particular path stays $1/2$ as the ball example illustrates -- since you are treating them as different then there are 2 paths available still, and nothing changes from your original setup.

You said they were same in 1 place but treated them different in the end, and you have to be consistent to get reasonable results, you cannot change the model midway...

Answer 2

You claim that these two paths are distinct, although you do not say what makes them distinct: $$ 2 \text{ then } 2 \tag A $$ $$ 2 \text{ then } 2 \tag B $$

Now you choose "$2$" (that is, you choose something you call "$2$," whatever that means to you). What is the probability that you chose the first path? It cannot be $2/2$ if the paths actually are distinct paths. If there are two distinct things both called "$2$," only one of these "$2$"s can be the first object on the first path, and you have only a $1/2$ chance to choose it as the first object; the other $1/2$ of the time you'll choose your other "$2$".

Alternatively, if the objects really are so interchangeable that picking either one of them will put you on the first path, and picking either one of them will put you on the second path, then as soon as you have picked an object you are on the first path and you are on the second path, that is, those two paths are actually the same path.

You have to decide which alternative interpretation you mean and stick with it. If you change the meanings of the words in your problem in the middle of the solution then you can very easily get wrong answers.