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Series XL Letter M episode 5: Maths - A Miscalculation?

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watermelemon
1192994.  Wed May 25, 2016 6:52 am Reply with quote

In this episode, Stephen got all four panel members, as well as himself, to shuffle a deck of cards each and place all five decks of cards into one bag. Then, all five pulled a card out one at a time from the bag, and everyone mysteriously pulled out a 6 of clubs. It was then stated that the chances of that happening are about 1 in 2 billion. Correct me if I am wrong, but I believe this may be incorrect, as the answer I found is approximately 1 in 10 billion.

We have 5 decks of 52 cards. Each deck has one 6♣.

Probability of pulling out a 6♣: 5/(52x5) = 5/260

We pull out 5 cards out of the bag, one at a time, and not putting them back once they're pulled out.

1st round: 5/260 (1/52)

For the second card, we have already pulled out one 6♣, so now there are only four in the bag. There is also now one less card in the bag.

2nd round: (5-1)/(260-1) = 4/259

Same for the third card: we have one less 6♣ in the bag, and hence one less card in the bag, as well. If we precede in this manner for all five times we pull out a card, we get the following:

P(5 6♣) = (5/260)(4/259)x (3/258)x (2/257)x (1/256)
= 5 ! /(260x259x258x257x256)
~= 1.0498 x 10^(-10)
...which is about 1 in 10 billion.

Again, if I calculated this incorrectly, please let me know.

 
Jenny
1193046.  Wed May 25, 2016 1:38 pm Reply with quote

Hi watermelemon and welcome to the forums :-)

We do have some mathematicians lying around here somewhere... paging Zziggy!

 
14-11-2014
1193099.  Thu May 26, 2016 6:04 am Reply with quote

Quote:
Probability of pulling out a 6♣: 5/(52x5) = 5/260

Was it a condition or prediction that the first card has to be 6♣? If not, FWIW, then your calculation is wrong (too?), because the first card could have been any card.

ISRT that mr Fry didn't predict that the first card would be 6♣, so you may have to leave out your first (5/260).

 
watermelemon
1193208.  Thu May 26, 2016 9:16 pm Reply with quote

Quote:
Was it a condition or prediction that the first card has to be 6♣?


The probability calculated was the probability of having five decks of cards in a bag, pulling out five cards in succession from that bag without replacing them, and having all five cards be 6♣. So yes, the first card, and each card, has to be a 6♣.

Quote:
ISRT that mr Fry didn't predict that the first card would be 6♣, so you may have to leave out your first (5/260).


I will have to check the episode, again, to make sure, but I am fairly certain that the calculation was based off all five cards being pulled out being 6♣, including the one that Stephen Fry pulled out.

 
PDR
1193211.  Fri May 27, 2016 2:10 am Reply with quote

I think the point 14 is making is that the requirement is for cards 2-6 to be the same as card 1, but card 1 could be anything.

Thus the first probability would be 1 rather than 1/52

Thus the final probability is

4 ! /(259x258x257x256) = ~5.5x10^(-9)

[one in five billion]

PDR

 
Zziggy
1193590.  Mon May 30, 2016 5:15 pm Reply with quote

Jenny wrote:
We do have some mathematicians lying around here somewhere... paging Zziggy!

Oh I only just saw this ... the three of you seem to have it covered though, I don't see anything immediately wrong here. I agree with PDR's 1 in 5.5 billion and watermelemon's 1 in 10 billion (not having seen the episode recently I can't tell you which is correct).

The maths elf must have been having a bad day ...

 
PDR
1193599.  Mon May 30, 2016 5:34 pm Reply with quote

Of course the actual probability was 1 in 1, because they cheated.

PDR

 
WordLover
1193617.  Tue May 31, 2016 2:28 am Reply with quote

.....


Last edited by WordLover on Thu Sep 15, 2016 9:23 am; edited 1 time in total

 
RLDavies
1194618.  Sun Jun 12, 2016 8:18 am Reply with quote

Just chipping in to say that it's a long-standing tradition in magic to misquote the odds of whatever miracle has just been pulled off.

Of course, the tradition is to make the result seem even less likely than it really is, not more likely. I suppose the whole thing just goes to prove that Mr Fry is not a proper magician (honorary Magic Circle membership notwithstanding).

 
CharliesDragon
1194703.  Sun Jun 12, 2016 10:35 pm Reply with quote

I just watched the episode (before seeing this thread) and it is not specified what card the first one might be. The first card can be any card, the important thing is that the following four is the same.

 

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