### Author Topic: Probability quiz  (Read 538 times)

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#### kav

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##### Probability quiz
« on: November 07, 2017, 01:14:57 PM »
There are 2 teachers teaching in two different classes, each class has 50 children.
Both teachers say that they will make a secret drawing each day and they will examine the student who's name is drawn.

After a year (roughly 300 drawings) we have these results:
Teacher A has drawn and examined the same student each and every day.
Teacher B has examined all 50 students over the year.

Some allegations have been made that teacher A hasn't been doing fair draws. To which he replied: "It's just the probabilities stupid! Every 300 draws sequence has the same chance. Picking the same student every day is no different than any other sequence. I'm a suspect for unfair draws as the next teacher. Thinking otherwise is gambler's fallacy."

What do you think?
« Last Edit: November 08, 2017, 12:36:39 AM by kav »

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#### Reyth

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##### Re: Probability quiz
« Reply #1 on: November 07, 2017, 02:03:44 PM »
Teacher A has just thrown out an entire branch of statistics called Cumulative Probability, with his first three statements:

Although his last two statements at the end make this a trick question imo because they are correct in both branches of statistics.

This kind of thinking where we examine each point that is made and notice the different meaning in them, has helped me to better appreciate Real's posts; often times now, I find great nuggets of wisdom tucked in between the banality.

You know this may be slightly off topic but this thread reminds me of the post where the number 18 came up, what was it, 11 times in a row?

I just now thought that if after 6 in a row, I started a positive D'Alembert for 5 consecutive wins, wouldn't that be grounds for the casino accusing me of cheating and refuse to pay me?
« Last Edit: November 07, 2017, 02:24:43 PM by Reyth »

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#### palestis

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##### Re: Probability quiz
« Reply #2 on: November 07, 2017, 10:06:40 PM »
Both situations under A and B are the opposite extremes. Though theoretically both situations are equally likely just as any other combination among the thousands possible,  realistically you will never see any of the 2 situations in your lifetime.
This is particularly important if you make a bet on either the A or B situation. You will lose your bet.
On the other hand, you will make a lot of money by betting hundreds of times that neither A or B will happen. Not necessarily a specific sequence, but any sequence other than A or B.
And that's the rational as it applies in roulette. ( though in more restricted choices).
You make several bets counting on the premise that the rare and unthinkable will not happen, during your bets. What is rare is up to the system designer to determine.
« Last Edit: November 07, 2017, 10:39:14 PM by palestis »

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#### kav

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##### Re: Probability quiz
« Reply #3 on: November 08, 2017, 12:44:12 AM »
Palestis,

Why do you consider an extreme even picking up all 50 names in 300 draws? 300 draws are 6x50name cycles. Or 6x37= 222 spins if we want to translate it onto roulette terms. Why is is almost impossible that all the possible outcomes appear?

The question is: is the A sequence less probable than any one SINGLE 300 draw sequence?
Of course you will make a lot of money if you bet against sequence A but this is because there are millions other possible sequences besides A. The question is if A is less likely versus any other sequence one-on-one.

#### Reyth

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##### Re: Probability quiz
« Reply #4 on: November 08, 2017, 03:25:56 AM »
Ya I misunderstood the question too -- not because it was unclear, but because my mind just "assumed" it was the polar opposite for Teacher B.

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#### kav

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##### Re: Probability quiz
« Reply #5 on: November 08, 2017, 12:12:06 PM »
Thanks for your comment Reyth,
It is probably my fault that one sees the question a "versus" question. It is not. Teacher B could have any kind of draw sequence. My point is that although many other sequences will look "normal" they are just ONE sequence out of millions of draw sequences, so THEORETICALLY a "normal" sequence has as little probability as the "crazy" sequence of teacher A.

Still, if the results of Teacher A are reason to doubt the fairness of his results, is open for debate. Because although all outcomes are theoretically possible, there are for example Chi square tests that can give estimation of the fairness of the "game" based on past results. So I'm not claiming that everything is fine with teacher A. I'm asking for opinions.

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#### palestis

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##### Re: Probability quiz
« Reply #6 on: November 08, 2017, 02:44:50 PM »
Palestis,

Why do you consider an extreme even picking up all 50 names in 300 draws? 300 draws are 6x50name cycles. Or 6x37= 222 spins if we want to translate it onto roulette terms. Why is it almost impossible that all the possible outcomes appear?

According to the terms of your question there is a draw EVERY DAY for a year or 300 days (excluding weekends and holidays I suppose), where the same student out of 50 gets picked.
That would be similar to 300 roulette spins where the same number out of 37 comes up.
Or if you want to use 222 spins, it means that the same number comes up 222 times in a row.
The probability of that happening (if you want to determine that number in advance), is
1/37x1/37x1/37................1/37,  222 times, which probably becomes one out of trillions in probability of happening.
On the other hand the situation B means that every number on the wheel appears in 300 spins or the 222 spins that you mention. Which sounds like a much more logical outcome.
Though before every spin, every number has the before the facts probability of 1/37 to appear, the 1 out trillions possibilities for that number to show up 222 times still works silently in the background. We may not be able to determine that number in advance, but that doesn't exclude any numbers to beat such a rare possibility.
The moral of the story is that every possibility will seek out its rightful place in the world of chances.
And that makes roulette beatable.
Though every number of the wheel  goes back to the drawing box, the power of each number to claim its rightful place cannot be overlooked. Only temporarily they lose that power, which we call it a variance.

« Last Edit: November 08, 2017, 03:16:00 PM by palestis »

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#### Reyth

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##### Re: Probability quiz
« Reply #7 on: November 08, 2017, 03:20:41 PM »
There is no common name for the "illions" that would be.  Just 11 times is in the quadrillions.

#### Jesper

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##### Re: Probability quiz
« Reply #8 on: December 04, 2017, 09:56:38 AM »
There is only on way to draw the same every day. Draw all only once has many ways. If we draw 300 numbers and each has been drawn once, it can be more than one order, but all the same number is just one of the way of all.

If we use just three numers

1,1,1  is only one of all.

123
231
321  we see has more.

« Last Edit: December 04, 2017, 10:00:59 AM by Jesper »

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#### Reyth

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##### Re: Probability quiz
« Reply #9 on: December 04, 2017, 10:39:15 PM »
But 123 is the same as 111 right?