# Thread: Best To Draw First Or Last?

1. ## Best To Draw First Or Last?

Hi, newbie non mathematician looking for an answer to a long debated question in angling.

When fishing a competition we draw a random number (say 1-20) which will determine your fishing spot (peg) for the day. There is normally one or two spots considered better than the rest. The argument is always whether there is a statistical advantage towards getting one of these spots if you are the first to draw or would drawing last make no difference.

So here's the question, in terms of marbles in a bag. If you have 19 white and one black in a bag will the probability of drawing out the black marble be greater the earlier you draw or is it just as probable that the black marble will be the last one left in the bag?

For what its worth I believe the probability of getting the black marble is exactly the same whether you draw first or last.

Thanks in advance for any help and explanation.

2. ## Re: Best To Draw First Or Last?

If each competitor’s draw remains secret until all the balls have been drawn then each person has an equal probability of drawing the black ball, in this case 1 in 20 or 0.05.

To see why, think about the situation where all participants reach into the bag simultaneously and grab one ball at random. While their hands are in the bag, nobody knows who has the black ball and each participant has the same chance of holding it. The situation where one person draws at a time is essentially no different as long as their draw remains secret until all balls have been drawn.

3. ## The Following User Says Thank You to Con-Tester For This Useful Post:

JesperHP (01-19-2016)

4. ## Re: Best To Draw First Or Last?

Thanks. I understand the concept of all drawing at once but I don't get the reason for the draw to be kept secret. Knowing what other people have drawn cannot surely affect what the next person draws given that it is a blind draw.

My line of reasoning is that to begin with you have a 19-1 chance of NOT drawing the black marble. So the odds are against you drawing it. As each white marble is withdrawn the chances of not drawing the black one get smaller while the odds of drawing it get greater until, if not drawn there is a 1-1 chance of drawing it last. As I see it the odds against decrease as the odds for increase at the same rate.

I was hoping for possibly a numerical statistic to show that the odds of the black remaining undrawn after 19 draws is the same as drawing it when drawing first.

As I say, not a maths scholar so it is instinct that tells me the odds are the same.

5. ## Re: Best To Draw First Or Last?

Strictly speaking, the draw doesn’t need to be kept secret. The a priori probability will remain unchanged at 1/20. I added the element of secrecy so as to avoid confusion by conditional probability, which is the very thing that tends to make people think that drawing later is better—that is, people tend to think erroneously that after the first few draws there will be fewer white balls (because there is a much greater probability of drawing a white ball when drawing early), and therefore an increased chance of picking the black ball when drawing later.

Also, if the results are known after each draw, the draw can cease as soon as the black ball is drawn. But in this case, it is known that the next drawer’s probability of drawing the black ball increases for as long as only white balls have been drawn.

But your reasoning is entirely correct: On any given draw, the overall probability that the black ball will be drawn is the same because the increased probability of drawing it is precisely offset by the decreased probability that the black ball has not yet been drawn. For example, the probability that the sixth person will draw the black ball is the composite of:

P = Person 1 did not draw the black ball AND Person 2 did not draw the black ball AND Person 3 did not draw the black ball AND Person 4 did not draw the black ball AND Person 5 did not draw the black ball AND Person 6 draws the black ball.

Using your numbers, this formally reduces to:
P = (19/20)×(18/19)×(17/18)×(16/17)×(15/16)×(1/15) = 1/20, which is the same as for any other person in the draw.

The same principle applies to whichever person you consider.

6. ## Re: Best To Draw First Or Last?

Thanks for the explanation. Really appreciate your taking the time. And thanks for confirming what my limited maths knowledge (M101 at OU) and intuition told me.

I have realised since asking the question that this can also be demonstrated empirically by drawing repeatedly and logging the results. With enough iterations the black ball would be drawn in equal numbers at any point from 1-20.

Of course conditional probability plays no part as in reality the draw has to be completed each time in order for each angler to have a spot to fish from.

7. ## Re: Best To Draw First Or Last?

De nada. My pleasure.

Originally Posted by Neil of the Nene
I have realised since asking the question that this can also be demonstrated empirically by drawing repeatedly and logging the results. With enough iterations the black ball would be drawn in equal numbers at any point from 1-20.
Quite so. This approach is known as a “Monte Carlo simulation”, an approach that is used in assorted scenarios, typically when analytical methods are too complex or as independent confirmation of analytical results. The short of it is that probabilities are limiting proportions over a theoretically unlimited number of samples which the Monte Carlo method aims to mimic with a large but limited number of samples.

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