[CamguyPEI]

You answered this question in 2008, but only said it was "obvious":

https://www.askmehelpdesk.com/mathematics/probability-getting-ace-50-cards-if-2-cards-randomly-removed-52-a-207122.html

Can you please help clear up a difference of opinion within our group? Here
are two questions we need answered:

QUESTION 1: A group of 10 guys is at a poker game, and there is a box of
beer containing exactly 10 bottles of beer. There is only ONE Alpine in the
box of 10 (and of course, 9 others). One at a time, each guy selects a beer
at random (without looking, or by any other random method) in hopes of
retrieving the Alpine. Now.. we all agree that the odds of the first guy
getting the Alpine is 1-in-10. However, what are the odds that the LAST
BEER to be selected is the Alpine? Is it also 1-in-10? Or, because there
is a good possibility that the Alpine has already been taken, have the odds
changed?

Here is a little video to help illustrate this:
http://www.youtube.com/watch?v=nNWfkWu3D5Y

QUESTION 2: In a freshly shuffled deck of standard playing cards, what are
the odds that the first card dealt will be an ace? We all agree that the
answer to this is 4-in-52 (or 1-in-13). But now that that card has been
dealt (and remains face down and unknown), what are the odds that the NEXT
card is an ace? Is it also 1-in-13? Or have the odds changed? An what
about after 29 cards have been dealt out... what are the odds that the 30th
card will be an ace?

Here is a little video to help illustrate this:
http://www.youtube.com/watch?v=Dh2U_J5TLYY

We appreciate your well-educated help with this.

Thank you,
Darin Foulkes
member, Le Poker Gang


*****
FYI, here is the opinion from the self-proclaimed right-thinking individual in the group. If you have time to (or if you care to) give a response, we'd love to have it.

"I suppose it would have been good to tweek the questions a little bit better. All that we right-thinking individuals were saying in regards to the Alpine situation, for example, is that the last guy choosing has less that a 1 in 10 chance of getting the Alpine. It`s the same with scenario number two, we look at it from the perspective of the guy being the last in line in getting a card. Let me see if I can make this simpler.

From the Alpine`s perspective, it`s odds of being chosen are 1 in 10. It doesn`t really care who chooses it as long as someone does. It`s its only way of having fun with someone.

But let`s put 10 guys around a table, and hand out the bottles one at a time going clockwise. Now assume that it`s Daylight Savings Time (don`t go ah **** here) and we are living in the Martimes which puts us ahead of some places and behind other places. But in any event, we`re still in the right time zone, if you know what I mean.

So the guy at 1 o`clock (let`s assume that it`s Guy McIntosh) gets the first beer and the guy at two o`clock (I think you will agree here that if Guy McIntosh was the guy at 1 o`clock he can`t be the guy at 2 o`clock, in other words the odds have changed) is in line for the second beer and so on. After Guy gets a beer, the other guys chances of getting the Alpine diminish because Guy McIntosh, who hasn`t been at a card game in a very long time, has decided to go on a toot. Thus the odds of getting an Alpine after Guy has had a go, have diminished greatly. It`s simple math.

Àll seriousness aside, here`s another scenario.

Everyone whose ever played cards knows that the chances of getting a royal flush from a fresh deck are approx. 1 in 650,000 because there are only 4 combinations of a five card hand out of 2.5 million possible 5 card hands. The 2.5 million hands are calculated from a pristine deck, ie, one that still has 52 cards. You have a 1 in 650,000 chance of getting a royal flush from a fresh deck because you are getting the first 5 cards (or they can be taken at random from the deck). After you`ve been dealt 5 cards, the next player`s probabilty of getting a royal flush diminishes because he doesn`t have the benefit of a full deck. Your best chance of ever getting a royal flush are if you get the first 5 cards. That`s why guys who play Let it Ride at the casino try to sit in seat number 1. It increases their chances of getting the best hand possible.

My point is that the guy choosing the last bottle will not have a 1 and 10 chance of getting the Alpine although at the start he did. In the same vein, the 10th guy sitting at the table has less chance of getting the Ace of Hearts than the first guy sitting at the table. But like the Alpine, from the Ace of Heart`s point of view, it has a 1 in 52 chance of being picked by any one of those guys at any time. That is the subtle difference that you seem not to grasp (and your fellow supporters, I might add).

It appears that we hold different perspectives so that in the end we may both be right although I may be righter than you but that doesn`t mean that you`re wrong.

So that`s why I`m wondering if the math teacher will understand the difference of perspectives given the way in which the questions were framed.

I agree with you that if you lay out 52 cards, having given 13 cards to 4 people,and then point at one card, whether it be the 24th card or the 44th card, the odds are 1 in 13 that it will be an ace. But if you ask the question--what are the odds that the person sitting in position four will get an ace, after three cards have been dealt before, his chances have diminished. That`s why there is a difference in perspective. An Ace will be an Ace no matter where it is and its odds in relation to the other cards in the deck will remain the same until the cows come home (unless you`re living on a farm, in which case you should refrain from reflecting on odds because the cows will do it to you every time).

So I would love to take your bet but the questions unfortunately are slanted in your favour. Perhaps you could send my email to the Math department and have the expert consider the clarified scenarios."

[QLP]

Sorry, I meant to do this but didn't have time.
This may help:

http://www.kibble.net/magic/magic10.php

[ebaines]

Your "right thinking" friend is wrong. The odds are the same for all players as long as the selections are truly random. Let's look at the Aplne example:

1. The probability of plater 1 "winning" is 1/10 - no dispute there. Which means the odds of him losing are 9/10.

2. For player 2 to win two things must occur: player 1 must lose and player 2 must get the Alpine out of the remaining 9 bottles. So the odds of him winning are:



Note that if player 2 got a chance to play (because player 1 already lost), then the probability of player 2 losing is 1 - 1/9 = 8/9.

3. For player 3 the probability of winning is equal to the probability that player 1 loses times the probability that player 2 loses times the probability that player 3 selects the one Alpine out of the 8 remaining bottles:



and so on. For the very last player, his odds are also 1//10:



As for why players may like to sit in seat 1: it has nothing to do with improving odds of getting particular cards but rather is psychological. - in playing Blackjack the player is seat 1 may feel more in "control" because the decisions he makes to take a card or hold will affect which cards the next players get. But it doesn't change the probabilioty of anyone getting a particular card. If there was truly a mathematical advantage to sitting is seat 1 there would be gun fights over it! One way to see it's not true - if the odds of getting a particular "good" card are better in seat 1 then by the same logic the odds of getting a particular bad card must also be higher for seat 1. The cards have no way of "knowing" whether they are considered "good" or "bad" by the humans around the table. If you believe that an ace is most likely to go seat 1 then you must also believe that a 2 is more likely to go to seat 1. So it should be clear that there is no difference in the odds for pulling a particular card no matter where you sit as long as the rules for dealing cards are independent for all players.