"Spelling Bee" odds on TPIR?

[BrentW]

Hey there all:

I searched this topic, but was unable to find anything remotely close to it.  I don't know if this is "spoilerish" but I can't get the damn Spoiler tags to work, so I am just going to use white letters.  HIGHLIGHT if you want to read....it doesn't spoil much, but I know some folks are spoiler-phobic.  

MINOR SPOILER:

[color=\"#FFFFFF\"]7 years after I got on TPIR, my little sister went out to it for the taping yesterday (Monday), after I told her exactly what to do.  I coached her a LOT.  I know many people say you can't coach someone onto a game show, but I don't think tht is true.  She was with me when I got on, so I told her what to do, how to act, what to say to the Producers, and I made her some shirts that I used when i got on, but just replaced "Bob" with "Drew" (hey, why not?  everyone who was there when I got on is now gone!  LOL) ....... Well, SHE GOT ON, BITCHES!!!!!!!!!  Not only that but she was one of the FIRST FOUR contetstants, and won the first IUFB to get on stage and play Spelling Bee!  She missed one of the 3 prizes and thus had 4 cards...  I don't know the result or if she won the Big Wheel/Showcase, but I WILL say that she told me she gets lots of TV time, and is on a LOT of the show.  :-)   So I am hoping for the best.  Her name is Lori, and it airs on Nov 10th.  She's got red hair, and porcelain white skin with a "Punch A Bunch" shirt on.  Her name changed to "Staifer" from "Wolgamott" since she is married.  LOOK FOR HER.  Apparently there is also a SPECIAL GUEST as well, but I do not know who it is.  She won't even tell me all of what happened to her!  LOL [/color]

OK, so now that the spoiler related stuff is out fo the way, I am curious: what are the odds of WINNING Spelling Bee with all 5 cards?  And then what do those odds look like if you have only 4 cards?  I know there have to be some geniuses who can do some permutations and computations on this board.  For your reference, there are 30 cards on the board, and there are

- 11 C's
- 11 A's
-  6 R's.
-  and two automatic win cards that spell "CAR".  

Any HELP would be appreciated.  I would assume this is easy to do, but I forget how to set up permutations and such.

[TLEberle]

There are 142,506 ways to pick five cards out of thirty. If one is a car, there are 23,751 ways to choose four cards out of the remaining twenty-nine. (since the car is already there, it makes no difference if the other four cards have CAR in there, or even the second CAR card.)  Assuming you don't find a car among your grouping, these are the combinations that have C-A-R in there;

CCCAR
CCAAR
CCARR
CAAAR
CAARR
CARRR

Figure out how many different ways you can make each winning combination, and subtract that from 30 C 5 (142,506) and ta-da, there's your P(win).

[chad1m]

I am not a bitch.

/And if I was, I'm not your bitch.

[clemon79]

[quote name=\'chad1m\' post=\'228479\' date=\'Oct 13 2009, 10:29 PM\']I am not a bitch.[/quote]
Approves

/was very happy with the winner this season

[Steve McClellan]

I'm a sucker for a math problem. Here's what I did.

To lose, you must lack a CAR card and you must lack at least one of C, A, and R. Ergo, the final probability of losing equation will be:

P(lose) = P(no CAR card) * P(no C, A, or R, given no CAR)

To not have a CAR card, you must miss them five times. At the start, there are 28 non-CARs out of 30, then (if you miss the first) 27 out of 29, etc. To find the chance of missing all five times, multiply:

P(no CAR card) = (28/30) * (27/29) * (26/28) * (25/27) * (24/26) = 11 793 600/17 100 720 = 20/29 (about .690)

Finding the probability of missing a C, A, or R is a bit more involved, but uses the same form of equation. Since we already took into account the chance of finding a CAR card, we deal with only the 28 remaining cards from here on.

P(no C) = (17/28) * ... * (13/24) = 17/270 (about .063)
P(no A) = P(no C) = 17/270
P(no R) = 209/780 (about .268)

Now comes the not-immediately-obvious part. Anytime you add probabilities, you have to check whether any situation would meet two or more of the conditions, and compensate for the fact that you would be counting it twice. In this case, five Cs would be counted under no A and no R; five As and five Rs give you similar issues. Since they're counted twice, subtract them once to correct.

P(no C, A, or R, given no CAR) = 17/270 + 17/270 + 209/780 - 11/2340 - 11/2340 - 1/16380 = 1889/4914 (about .384)

So the final equation looks like this:

P(lose) = P(no CAR card) * P(no C, A, or R, given no CAR)
P(lose) = 20/29 * 1889/4914
P(lose) = 18890/71253 (about .2651)

P(win) = 1 - P(lose) = 52363/71253 (about .7349)

Ergo, unless I have royally screwed something up, there is a roughly 73.49% chance that a player with five cards will win, ignoring the possibility of quitting early and taking cash. By the same math, the chances of winning are:

5 cards: 52363/71253 = 73.49%
4 cards: 1067/1827 = 58.40%
3 cards: 151/406 = 37.19%
2 cards: 19/145 = 13.10%

[NickS]

[quote name=\'clemon79\' post=\'228480\' date=\'Oct 14 2009, 12:39 AM\'][quote name=\'chad1m\' post=\'228479\' date=\'Oct 13 2009, 10:29 PM\']I am not a bitch.[/quote]
Approves
[/quote]

Disagrees.

/Interesting to see how those five guys turned out

[Mr. Armadillo]

[quote name=\'Steve McClellan\' post=\'228483\' date=\'Oct 14 2009, 01:13 AM\']Ergo, unless I have royally screwed something up, there is a roughly 73.49% chance that a player with five cards will win, ignoring the possibility of quitting early and taking cash. By the same math, the chances of winning are:

5 cards: 52363/71253 = 73.49%
4 cards: 1067/1827 = 58.40%
3 cards: 151/406 = 37.19%
2 cards: 19/145 = 13.10%[/quote]
The math has been done on this forum before, and you've got the same percentages that I remember seeing from that post (which I have referenced when the question came up on golden-road more than once), so I'm pretty sure you've done it right.

So, Brent, with four cards, your sister was actually a small favorite to win the car.

[Jeremy Nelson]

[quote name=\'clemon79\' post=\'228480\' date=\'Oct 14 2009, 12:39 AM\'][quote name=\'chad1m\' post=\'228479\' date=\'Oct 13 2009, 10:29 PM\']I am not a bitch.[/quote]
Approves

/was very happy with the winner this season
[/quote]
Agreed...although either of the two finalists was fine with me, cause they were both really good.

[clemon79]

[quote name=\'rollercoaster87\' post=\'228492\' date=\'Oct 14 2009, 09:18 AM\']Agreed...although either of the two finalists was fine with me, cause they were both really good.[/quote]
Exactly so. I actually had those two picked from Day One as the only two contestants with any competence. But one was clearly better than the other in the final service, and the outcome reflected that. But if the final service performances had been reversed, I would have totally been fine with the other one winning. Both were worthy champions.

/oh, and thanks, Fox, for spoiling it in the opening montage, you jackbags

[Jeremy Nelson]

[quote name=\'clemon79\' post=\'228495\' date=\'Oct 14 2009, 12:04 PM\']/oh, and thanks, Fox, for spoiling it in the opening montage, you jackbags[/quote]
Glad I missed the first hour.

I liked the fact, though, that they got the final 3 and the finals done in a 2 hour finale. Usually, it would be three separate one-hour shows, along with all of the "restaurant redesign" stuff that I really didn't care for. It was quite pleasant.

[BrentW]

[quote name=\'chad1m\' post=\'228479\' date=\'Oct 14 2009, 01:29 AM\']I am not a bitch.

/And if I was, I'm not your bitch.[/quote]


[quote name=\'clemon79\' post=\'228480\' date=\'Oct 14 2009, 01:39 AM\'][quote name=\'chad1m\' post=\'228479\' date=\'Oct 13 2009, 10:29 PM\']I am not a bitch.[/quote]
Approves

/was very happy with the winner this season
[/quote]


I'm guessing this is an inside joke concerning Hell's Kitchen or something, a show I do not watch.  When I'm excited, I tend to say "bitches" a lot.  :-)

[quote name=\'Steve McClellan\' post=\'228483\' date=\'Oct 14 2009, 02:13 AM\']I'm a sucker for a math problem. Here's what I did.

To lose, you must lack a CAR card and you must lack at least one of C, A, and R. Ergo, the final probability of losing equation will be:

P(lose) = P(no CAR card) * P(no C, A, or R, given no CAR)

To not have a CAR card, you must miss them five times. At the start, there are 28 non-CARs out of 30, then (if you miss the first) 27 out of 29, etc. To find the chance of missing all five times, multiply:

P(no CAR card) = (28/30) * (27/29) * (26/28) * (25/27) * (24/26) = 11 793 600/17 100 720 = 20/29 (about .690)

Finding the probability of missing a C, A, or R is a bit more involved, but uses the same form of equation. Since we already took into account the chance of finding a CAR card, we deal with only the 28 remaining cards from here on.

P(no C) = (17/28) * ... * (13/24) = 17/270 (about .063)
P(no A) = P(no C) = 17/270
P(no R) = 209/780 (about .268)

Now comes the not-immediately-obvious part. Anytime you add probabilities, you have to check whether any situation would meet two or more of the conditions, and compensate for the fact that you would be counting it twice. In this case, five Cs would be counted under no A and no R; five As and five Rs give you similar issues. Since they're counted twice, subtract them once to correct.

P(no C, A, or R, given no CAR) = 17/270 + 17/270 + 209/780 - 11/2340 - 11/2340 - 1/16380 = 1889/4914 (about .384)

So the final equation looks like this:

P(lose) = P(no CAR card) * P(no C, A, or R, given no CAR)
P(lose) = 20/29 * 1889/4914
P(lose) = 18890/71253 (about .2651)

P(win) = 1 - P(lose) = 52363/71253 (about .7349)

Ergo, unless I have royally screwed something up, there is a roughly 73.49% chance that a player with five cards will win, ignoring the possibility of quitting early and taking cash. By the same math, the chances of winning are:

5 cards: 52363/71253 = 73.49%
4 cards: 1067/1827 = 58.40%
3 cards: 151/406 = 37.19%
2 cards: 19/145 = 13.10%[/quote]

THANK YOU STEVE!  Much appreciated.  I knew the odds were higher than they "look," so to speak, but it is nice to see it written out.  And I can actually follow it!

Regards,
Brent

[clemon79]

[quote name=\'BrentW\' post=\'228510\' date=\'Oct 14 2009, 03:12 PM\']I'm guessing this is an inside joke concerning Hell's Kitchen or something, a show I do not watch.[/quote]
See "nominees for elimination".

When I'm excited, I tend to say "bitches" a lot.  :-)

Suggests you rethink that

[BrentW]

[quote name=\'clemon79\' post=\'228512\' date=\'Oct 14 2009, 06:25 PM\'][quote name=\'BrentW\' post=\'228510\' date=\'Oct 14 2009, 03:12 PM\']I'm guessing this is an inside joke concerning Hell's Kitchen or something, a show I do not watch.[/quote]
See "nominees for elimination".

When I'm excited, I tend to say "bitches" a lot.  :-)

Suggests you rethink that
[/quote]

Noted for consideration.  However, I'm not in the Hilary Duff camp of thinking every random utterance of a non-PC term is cause for a bloody public service announcement (or, apparently, for a few posts noting such).  Nevertheless, I know what getting into a discussion about this issue will get me on these boards (ie: loads of flaming, eye-rolling, and public floggings via postings).  So, that said,....

Thank you to Steve and Armadillo for contributions to the ACTUAL point to this thread.  It can be closed now, or left to die a slow death rolling off the front page...  :-)

That was SO gay,
Brent  :-)

[NickS]

[quote name=\'BrentW\' post=\'228515\' date=\'Oct 14 2009, 06:14 PM\']Noted for consideration.  However, I'm not in the Hilary Duff camp of thinking every random utterance of a non-PC term is cause for a bloody public service announcement (or, apparently, for a few posts noting such).  Nevertheless, I know what getting into a discussion about this issue will get me on these boards (ie: loads of flaming, eye-rolling, and public floggings via postings).  So, that said,....

Thank you to Steve and Armadillo for contributions to the ACTUAL point to this thread.  It can be closed now, or left to die a slow death rolling off the front page...  :-)

That was SO gay,
Brent  :-)[/quote]

Brent, with all due respect - if it wasn't for me looking at the spoilered part, I questioned what Chad said.  Then, going back over it - you opened this can of worms.  If you didn't put the "bitches" up, I bet you would have had a different response.

ObTPiR: Since I'm not the one to ask about probabilites questions, good luck to your sister on the show!