Finding the Right Revisiting a Two-Year-Old Problem

[veronica1999]

I solved this attached problem two years ago and now I am starting to think the solution is wrong.
And the problem is flawed. Can someone please help clarify my understanding?

 

[CaptainBlack]

I solved this attached problem two years ago and now I am starting to think the solution is wrong.
And the problem is flawed. Can someone please help clarify my understanding?

Your original solution is correct.

You can think of the darts as thrown one after the other so the probability of an odd outcome is the sum of the probabilities that the first is odd and the second even and that the first is even and the second odd.

CB

 

[Sudharaka]

I solved this attached problem two years ago and now I am starting to think the solution is wrong.
And the problem is flawed. Can someone please help clarify my understanding?

Hi veronica1999,

As CaptainBlack had already told, your first answer is correct. Let me explain how I think about it.

What are the events and outcomes of this experiment? The problem says,

The probability that a dart will hit a given region is proportional to the area of the region.

Outcome 1: A dart hitting a region numbered "1".

Outcome 2: A dart hitting a region numbered "2".

We know the probabilities of these outcomes.

Then there are various events that could occur, for example the first dart hitting the outer region numbered 2 and the second dart hitting a inner region numbered 1. Out of these events we are interested about the events that have sum=3.

Suppose if you throw the two darts at the same time, and moreover they hit the board at the same instance. Then what will be the outcomes of this experiment? We may be able to define outcomes such as,

Outcome 1: The darts achieving a sum of "3".

Outcome 2: The darts not achieving a sum of "3".

However do we know the probabilities of these outcomes? No. You are given only the probabilities that a dart hitting a certain region, not the probabilities of the outcomes that occur when the two darts hit the board together.

What you have done wrong in the second method is that you have taken your outcomes as,

Outcome 1: A dart hitting a region numbered "1".

Outcome 2: A dart hitting a region numbered "2".

and done some calculations. But finally you have thought about the outcomes again as,

Outcome 1: The darts achieving a sum of "3".

Outcome 2: The darts not achieving a sum of "3".

The summary is, you can throw the darts at once, however the probabilities of the outcomes that you will get, you don't know.

I hope my explanation is clear enough to clarify your doubts.

Kind Regards,
Sudharaka.