Probability of guessing the colors of a deck of cards correctly
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10 years ago when I was about 15 I sat down with a deck of shuffled cards and tried to guess if the next card in the deck would be red or black. In sequence I guessed 36 cards correctly as red or black, at that point my older bother came in and took told me what I was doing was stupid and I stopped being able to guess correctly. I would love to know what the probability of guessing 36 cards correctly in sequence is? Thank you
probability-theory
migrated from mathoverflow.net Dec 10 '14 at 12:39
This question came from our site for professional mathematicians.
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up vote
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10 years ago when I was about 15 I sat down with a deck of shuffled cards and tried to guess if the next card in the deck would be red or black. In sequence I guessed 36 cards correctly as red or black, at that point my older bother came in and took told me what I was doing was stupid and I stopped being able to guess correctly. I would love to know what the probability of guessing 36 cards correctly in sequence is? Thank you
probability-theory
migrated from mathoverflow.net Dec 10 '14 at 12:39
This question came from our site for professional mathematicians.
2
A guess of "red or black" has a high probability of being correct. You could succeed $36$ times in a row.
– Ross Millikan
Dec 10 '14 at 16:21
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
10 years ago when I was about 15 I sat down with a deck of shuffled cards and tried to guess if the next card in the deck would be red or black. In sequence I guessed 36 cards correctly as red or black, at that point my older bother came in and took told me what I was doing was stupid and I stopped being able to guess correctly. I would love to know what the probability of guessing 36 cards correctly in sequence is? Thank you
probability-theory
10 years ago when I was about 15 I sat down with a deck of shuffled cards and tried to guess if the next card in the deck would be red or black. In sequence I guessed 36 cards correctly as red or black, at that point my older bother came in and took told me what I was doing was stupid and I stopped being able to guess correctly. I would love to know what the probability of guessing 36 cards correctly in sequence is? Thank you
probability-theory
probability-theory
asked Dec 10 '14 at 10:46
Adam Sunderland
migrated from mathoverflow.net Dec 10 '14 at 12:39
This question came from our site for professional mathematicians.
migrated from mathoverflow.net Dec 10 '14 at 12:39
This question came from our site for professional mathematicians.
2
A guess of "red or black" has a high probability of being correct. You could succeed $36$ times in a row.
– Ross Millikan
Dec 10 '14 at 16:21
add a comment |
2
A guess of "red or black" has a high probability of being correct. You could succeed $36$ times in a row.
– Ross Millikan
Dec 10 '14 at 16:21
2
2
A guess of "red or black" has a high probability of being correct. You could succeed $36$ times in a row.
– Ross Millikan
Dec 10 '14 at 16:21
A guess of "red or black" has a high probability of being correct. You could succeed $36$ times in a row.
– Ross Millikan
Dec 10 '14 at 16:21
add a comment |
2 Answers
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If you just guess red or black with a 50/50 chance, without taking into account that memory of the previous cards can make a guess more accurate, then the probability of guessing 36 cards in a row correctly is:
$$left(frac 12right)^{36}$$
For comparison, if all 7.125 billion people in the world were to attempt this, the probability that anyone in the world would succeed is
$$left(frac 12right)^{36} ~~times~~ 7.125 times 10^9 = 10.4%$$
Which makes your story somewhat unbelievable.
answered Dec 10 '14 at 16:30
DanielV
17.8k42754
Well, if everybody tried this ten times in their lives, probably someone would be able to report success.
– Hagen von Eitzen
Dec 10 '14 at 16:32
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If you're aiming to guess the first 36 cards right with as high a probability as possible, then the best you can do is to choose some sequence of guesses with 18 reds and 18 blacks, and all those sequences are equally likely.
Each of them has probability
$frac{(26times 25times 24timesdotstimes9)^2}{52times51times50timesdotstimes17}$
$=frac{26!times26!times16!}{52!times8!times8!}$
$approx 2.595times 10^{-11}$.
So, the chances are about 1 in 40 billion.
(Assuming the pack of cards is shuffled)
It's easier than guessing 36 coin-tosses correctly, which would have probability $2^{-36}$ which is about 1 in 70 billion.
answered Dec 10 '14 at 16:11
James Martin
20716
add a comment |
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2 Answers
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2 Answers
2
active
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active
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active
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up vote
1
down vote
If you just guess red or black with a 50/50 chance, without taking into account that memory of the previous cards can make a guess more accurate, then the probability of guessing 36 cards in a row correctly is:
$$left(frac 12right)^{36}$$
For comparison, if all 7.125 billion people in the world were to attempt this, the probability that anyone in the world would succeed is
$$left(frac 12right)^{36} ~~times~~ 7.125 times 10^9 = 10.4%$$
Which makes your story somewhat unbelievable.
answered Dec 10 '14 at 16:30
DanielV
17.8k42754
Well, if everybody tried this ten times in their lives, probably someone would be able to report success.
– Hagen von Eitzen
Dec 10 '14 at 16:32
add a comment |
up vote
1
down vote
If you just guess red or black with a 50/50 chance, without taking into account that memory of the previous cards can make a guess more accurate, then the probability of guessing 36 cards in a row correctly is:
$$left(frac 12right)^{36}$$
For comparison, if all 7.125 billion people in the world were to attempt this, the probability that anyone in the world would succeed is
$$left(frac 12right)^{36} ~~times~~ 7.125 times 10^9 = 10.4%$$
Which makes your story somewhat unbelievable.
answered Dec 10 '14 at 16:30
DanielV
17.8k42754
Well, if everybody tried this ten times in their lives, probably someone would be able to report success.
– Hagen von Eitzen
Dec 10 '14 at 16:32
add a comment |
up vote
1
down vote
up vote
1
down vote
If you just guess red or black with a 50/50 chance, without taking into account that memory of the previous cards can make a guess more accurate, then the probability of guessing 36 cards in a row correctly is:
$$left(frac 12right)^{36}$$
For comparison, if all 7.125 billion people in the world were to attempt this, the probability that anyone in the world would succeed is
$$left(frac 12right)^{36} ~~times~~ 7.125 times 10^9 = 10.4%$$
Which makes your story somewhat unbelievable.
answered Dec 10 '14 at 16:30
DanielV
17.8k42754
If you just guess red or black with a 50/50 chance, without taking into account that memory of the previous cards can make a guess more accurate, then the probability of guessing 36 cards in a row correctly is:
$$left(frac 12right)^{36}$$
For comparison, if all 7.125 billion people in the world were to attempt this, the probability that anyone in the world would succeed is
$$left(frac 12right)^{36} ~~times~~ 7.125 times 10^9 = 10.4%$$
Which makes your story somewhat unbelievable.
answered Dec 10 '14 at 16:30
DanielV
17.8k42754
edited Dec 10 '14 at 16:38
answered Dec 10 '14 at 16:30
DanielV
17.8k42754
answered Dec 10 '14 at 16:30
DanielV
17.8k42754
answered Dec 10 '14 at 16:30
DanielV
17.8k42754
17.8k42754
Well, if everybody tried this ten times in their lives, probably someone would be able to report success.
– Hagen von Eitzen
Dec 10 '14 at 16:32
add a comment |
Well, if everybody tried this ten times in their lives, probably someone would be able to report success.
– Hagen von Eitzen
Dec 10 '14 at 16:32
Well, if everybody tried this ten times in their lives, probably someone would be able to report success.
– Hagen von Eitzen
Dec 10 '14 at 16:32
Well, if everybody tried this ten times in their lives, probably someone would be able to report success.
– Hagen von Eitzen
Dec 10 '14 at 16:32
add a comment |
up vote
1
down vote
If you're aiming to guess the first 36 cards right with as high a probability as possible, then the best you can do is to choose some sequence of guesses with 18 reds and 18 blacks, and all those sequences are equally likely.
Each of them has probability
$frac{(26times 25times 24timesdotstimes9)^2}{52times51times50timesdotstimes17}$
$=frac{26!times26!times16!}{52!times8!times8!}$
$approx 2.595times 10^{-11}$.
So, the chances are about 1 in 40 billion.
(Assuming the pack of cards is shuffled)
It's easier than guessing 36 coin-tosses correctly, which would have probability $2^{-36}$ which is about 1 in 70 billion.
answered Dec 10 '14 at 16:11
James Martin
20716
add a comment |
up vote
1
down vote
If you're aiming to guess the first 36 cards right with as high a probability as possible, then the best you can do is to choose some sequence of guesses with 18 reds and 18 blacks, and all those sequences are equally likely.
Each of them has probability
$frac{(26times 25times 24timesdotstimes9)^2}{52times51times50timesdotstimes17}$
$=frac{26!times26!times16!}{52!times8!times8!}$
$approx 2.595times 10^{-11}$.
So, the chances are about 1 in 40 billion.
(Assuming the pack of cards is shuffled)
It's easier than guessing 36 coin-tosses correctly, which would have probability $2^{-36}$ which is about 1 in 70 billion.
answered Dec 10 '14 at 16:11
James Martin
20716
add a comment |
up vote
1
down vote
up vote
1
down vote
If you're aiming to guess the first 36 cards right with as high a probability as possible, then the best you can do is to choose some sequence of guesses with 18 reds and 18 blacks, and all those sequences are equally likely.
Each of them has probability
$frac{(26times 25times 24timesdotstimes9)^2}{52times51times50timesdotstimes17}$
$=frac{26!times26!times16!}{52!times8!times8!}$
$approx 2.595times 10^{-11}$.
So, the chances are about 1 in 40 billion.
(Assuming the pack of cards is shuffled)
It's easier than guessing 36 coin-tosses correctly, which would have probability $2^{-36}$ which is about 1 in 70 billion.
answered Dec 10 '14 at 16:11
James Martin
20716
If you're aiming to guess the first 36 cards right with as high a probability as possible, then the best you can do is to choose some sequence of guesses with 18 reds and 18 blacks, and all those sequences are equally likely.
Each of them has probability
$frac{(26times 25times 24timesdotstimes9)^2}{52times51times50timesdotstimes17}$
$=frac{26!times26!times16!}{52!times8!times8!}$
$approx 2.595times 10^{-11}$.
So, the chances are about 1 in 40 billion.
(Assuming the pack of cards is shuffled)
It's easier than guessing 36 coin-tosses correctly, which would have probability $2^{-36}$ which is about 1 in 70 billion.
answered Dec 10 '14 at 16:11
James Martin
20716
edited Dec 11 '14 at 0:20
answered Dec 10 '14 at 16:11
James Martin
20716
answered Dec 10 '14 at 16:11
James Martin
20716
answered Dec 10 '14 at 16:11
James Martin
20716
20716
add a comment |
add a comment |
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2
A guess of "red or black" has a high probability of being correct. You could succeed $36$ times in a row.
– Ross Millikan
Dec 10 '14 at 16:21