Red / Black Cards. Puzzle

[jason_t]

[ QUOTE ]
I found it surprising -- in the sense that it's counter-intuitive. When I read over it, I had to stop and pause --then look at it again.

[/ QUOTE ]

I'm saying that the solution shows why it's not counterintuitive. 26c13 is much much bigger than 25c13. There are a lot more ways to draw all black cards than there are all red cards.

[SheetWise]

[ QUOTE ]
I'm saying that the solution shows why it's not counterintuitive.

[/ QUOTE ]
intuitive
adj 1: spontaneously derived from or prompted by a natural tendency; "an intuitive revulsion" 2: obtained through intuition rather than from reasoning or observation.

coun·ter·in·tu·i·tive
adj. Contrary to what intuition or common sense would indicate: “Scientists made clear what may at first seem counterintuitive, that the capacity to be pleasant toward a fellow creature is... hard work” (Natalie Angier).

[jason_t]

[ QUOTE ]
[ QUOTE ]
I'm saying that the solution shows why it's not counterintuitive.

[/ QUOTE ]
intuitive
adj 1: spontaneously derived from or prompted by a natural tendency; "an intuitive revulsion" 2: obtained through intuition rather than from reasoning or observation.

coun·ter·in·tu·i·tive
adj. Contrary to what intuition or common sense would indicate: “Scientists made clear what may at first seem counterintuitive, that the capacity to be pleasant toward a fellow creature is... hard work” (Natalie Angier).

[/ QUOTE ]

I'm telling you that your intuition was bad. Wrong. Incorrect. Not good. Terrible. Should have been different. Removing one card makes a huge difference because of how fast the number of ways of choosing k things from n things grows as k stays fixed but n increases. If you're at all familiar with these types of problems then your intuition would have been correct. Otherwise, if you aren't familiar with these types of problems then it's not fair to say you have any intuition about them.

[SheetWise]

It really was unfair of the dictionary to define "intuitive" as "obtained through intuition rather than from reasoning or observation." To fill in the blank ...

in·tu·i·tion
The act or faculty of knowing or sensing without the use of rational processes; immediate cognition.

Unlike you, I sometimes come across things that surprise me. Things that are true, yet they sound wrong. Sometimes it's just the way questions or problems are worded. I didn't say that it didn't make sense, or that I didn't understand it.

[PairTheBoard]

Let's see how close this method comes to the correct solution. You pull one card. Chance it's black is 26/51. You pull 25 cards, all the same color. Chance they're black is 26/27. 13 is about halfway in between so averaging the two probabilities you get about 74%. Hey, not too bad.

PairTheBoard

[jason_t]

[ QUOTE ]
It really was unfair of the dictionary to define "intuitive" as "obtained through intuition rather than from reasoning or observation." To fill in the blank ...

in·tu·i·tion
The act or faculty of knowing or sensing without the use of rational processes; immediate cognition.

Unlike you, I sometimes come across things that surprise me. Things that are true, yet they sound wrong. Sometimes it's just the way questions or problems are worded. I didn't say that it didn't make sense, or that I didn't understand it.

[/ QUOTE ]

I come across such things too. Here's one example: if 23 people are in a room then it's even money that two people in the room share the same birthday. That is shocking. That goes against everyday intuition.

But one has to have a certain amount of knowledge built up in an area before it's fair to say that something surprises them and that it goes against their intuition. For example, I could tell you that every function that is analytic on the entire complex plane and bounded is constant and unless you're a student of mathematics it is not fair to say to say that goes against your intuition. If you are a student of mathematics and you're encountering that statement for the first time, it's incredibly shocking and goes against intuition.

The other day I encountered this problem. If I had never heard the above statement about 23 people and birthdays, the answer to it would go against my intuition. But now my intuition has developed and the answer to that second problem isn't surprising to me.

I firmly believe, but could be wrong, that if one thinks they have intuition about this sort of problem (the red/black problem) then it should be in accordance with the solution and otherwise it's not fair to say that they have intuition about it. Either you are aware that the number of ways to choose to k things from n objects grows really fast or you don't. If you don't, how could you have any intuition about it? You have no natural sense about what is going on.

Finally, I sense by the tone of your last post that you might think that I think you're stupid and am being condescending. I don't think that at all and I am not trying to be. I'm not even in a position to judge that based on one conversation about one little problem. I hope clarifying my philosophy on intuition clears that up.

[punter11235]

I agree with you that the answer is not counter intuitive. (I am a student of computer science so there may be sth in it)

[SheetWise]

Thank you for your message. I see from your profile that you're a student. I actually do have a context to understand many of these problems -- but I graduated the University in 1976. During the near 30 years since then, raising four kids -- making a living and such, a lot of it gets blurry. Dealing with a computer revolution, birth of the Internet, a dozen OS's, a couple dozen languages, and such --- a lot of it was lost. I've been programming all this time, and even managed to survive as an independent in the gaming industry -- but, as you develop clients and relationships the work tends to go in the clients direction. It's amazing how much of what we learn we never use -- and I miss it -- which is why I'm now spending a lot of time on this site. Almost all of the work I do is in gaming -- but not cards (other than BJ, baccarat, etc), and usually not from the players perspective. Slot machines, parimutuels, sports, lotteries, etc. A lot of simulations. Even so, 2/3 raised my eyebrow -- before I reviewed it.

This is a great site, you and people like you contribute a lot -- and I thank you. I hope to get back up to speed, and participate more in the poker forums. In the meantime I'm bouncing around, reading, studying, trying some of your puzzles, and learning. I do thank you all.

[TimTimSalabim]

The probability of all black cards is 26/51 * 25/50 * ... * 14/39. The probability of all red is 25/51 * 24/50 * ... * 13/39. The denominators are identical in both cases, and the numerators are the same except for the first black (26) and the last red (13). Therefore, all black is 26/13 or twice as likely as all red.

[durron597]

What's interesting to me here is that if you have an arbitrary number of cards in the deck (x), and are pulling an arbitrary number of cards out of the deck (y), then the probability that they are all the majority color is:

x / (2x - y). If y = x/2, then we have x / (3x / 2) = 2/3. Thus the probability is 2/3 independent of deck size.