Calculating the Probability of Guessing the 51st Card Color

[blumfeld0]

This question is driving me crazy
It is a bayesian probability question
what is the probability of guessing the color of the 51st card correctly IF you have already drawn 50 cards and know exactly what colors each one cameout to be


thank you


khurram

 

[siddharth]

What are your ideas or thoughts on this problem? You need to show your work to get help.

 

[arunbg]

Just look at what colours the 51st and 52nd cards can turn out to be.
If they are of same colour you can guess the colour of 51st card correctly
(since you should have drawn all cards of the other colour earlier).
If they are of diff. colours you can easily find the probability that your guess is right. Multiply to get the probability.

 

[Curious3141]

As Arun has stated, you have to consider what colors the next to last and last cards ought to be. They can be similar (both blacks or both reds) or dissimilar (red, black or black, red). Find the probability of each case. Caution : the prob. of similar is not equal to dissimilar.

 

[blumfeld0]

I don't get it
the probability of him guessing the 51st card corrrectly is then 25/50 *26/52?
what is the probability of getting both blacks or red?
what is the probability of getting red, black and black red?
thanks
blumfeld

 

[Curious3141]

As a general result, for a set of N items (N even), with half of one sort and half of the other, I get the probability of guessing the last two right after being shown the first (N-2) as being :

p = \frac{3N-4}{4(N-1)}

 

[Curious3141]

I don't get it
the probability of him guessing the 51st card corrrectly is then 25/50 *26/52?
what is the probability of getting both blacks or red?
what is the probability of getting red, black and black red?
thanks
blumfeld

I'm afraid the answer isn't correct.

Take it in steps. I found it easiest to see the problem from a combinatorics perspective. What are the total number of ways you can arrange N/2 cards of one type and N/2 cards of the other to give a deck of N cards?

How many of those arrangements will have the same type of card for the last two?

How many will have one of each type for the last two?

Hence work out the probabilities of each scenario materialising.

Now for each of those scenarios, what is the probability of guessing the (N-1)th card correctly?

 

[balakrishnan_v]

you have 2 cards left.
Since you already know the colors of all other cards,you know the colors that the last 2 cards can take
each of the card can take one of the 4 possible colors and you know the color pair
C1 C1
C1 C2
.
.
.16 pairs


for C1,C1 C2C2 C3C3 C4C4 you will judge it perfectly(4 out of 16)
for any other color pairs(12 out of 16),there is a 1/2 probability of getting it right


So the net probability is 1/4*1+3/4*1/2=5/8

 

[Curious3141]

you have 2 cards left.
Since you already know the colors of all other cards,you know the colors that the last 2 cards can take
each of the card can take one of the 4 possible colors and you know the color pair
C1 C1
C1 C2
.
.
.16 pairs


for C1,C1 C2C2 C3C3 C4C4 you will judge it perfectly(4 out of 16)
for any other color pairs(12 out of 16),there is a 1/2 probability of getting it right


So the net probability is 1/4*1+3/4*1/2=5/8

I think you're mixing up 'suits' with 'colors'. A standard deck of cards has four suits (diamonds, hearts, spades and clubs) but only two colors (red and black).

 

[balakrishnan_v]

I did not know that.I just looked up the net and somewhere it was written cards have 4 colors.
In any case for a 2 color suite,I guess the answer would be 3/4

 

[Curious3141]

OK, here's how I did it. I prefer to derive a more general result here.

First of all, it's immediately obvious that in a deck of N cards (N is even) with N/2 of one color and N/2 of the other color, if all N/2 cards of one color have been drawn by the time the (N-2)th card has been drawn, there is no guesswork entailed in predicting the last two cards (prob. of correct guess = 1). Let's call this Scenario 1.

If, in contrast (Scenario 2), exactly (N/2 - 1) of one type and (N/2 - 1) of the other had been drawn by that time, the prob. of a correct guess is only half. So the overall probability has to be somewhere between half and one.

I found it difficult to predict the respective probabilities of Scenario 1 and 2 without using a combinatorics approach. With that approach, it becomes easy.

We're given a deck of N cards, well shuffled. Array them out in order.

The total number (N) of possible distinct arrangements of these N cards is N = \frac{N!}{({(\frac{N}{2})!)}^2}

In the case where the last two cards are dissimilar (Scenario 2), (N-2)/2 cards of each sort have to be arranged to fill the first (N-2) places .

The number of ways that this can happen is n_2 = 2\frac{(N-2)!}{({(\frac{N-2}{2})!)}^2} (The multiplication by two is to take into account both the (red, black) and (black, red) possibilities).

In Scenario 1, (N/2) cards of one color and (N-4)/2 cards of the other color have to be arranged.

The number of ways for Sc. 1 is n_1 = 2\frac{(N-2)!}{(\frac{N}{2})!(\frac{N-4}{2})!}

Again, the expression is multiplied by two to consider the cases where one color predominates, and vice versa.

The respective probabilities for Sc 1 and 2 are :

p_1 = \frac{n_1}{N} = \frac{N-2}{2(N-1)}
p_2 = \frac{n_2}{N} = \frac{N}{2(N-1)}

(Alternatively, you could simply take p_2 = 1 - p_1 but this way you can check your work).

If Scenario 1 materialises, the probability of guessing the last two cards right is one (no guesswork). If Scenario 2 materialises, the probability of correctly guessing the last two right is 1/2.

So the overall probability of a correct guess is p = (1)p_1 + \frac{1}{2}p_2 = \frac{3N-4}{4(N-1)}

For a deck of 52 cards, that gives p = \frac{38}{51}.

 

[Curious3141]

In any case for a 2 color suite,I guess the answer would be 3/4

This would only hold if the chances of getting two similar and dissimilar cards in the last two places were equal, which isn't the case. This is why I specifically cautioned the OP against assuming they were equal.

The answer is close to 3/4 though. This is because 52 is a fairly large number. The limit of p as N tends to infinity is 3/4.

 

[arunbg]

Nice job there Curious.

To find the probability of the card colours turning out a certain way, an easier approach would be to reverse the deck and deal two cards from the top. What is the probability that the two cards are of the same colour? Of diff. colours?
I haven't tried this ( too lazy ) but you ought to get the same answer shouldn't you ?

 

[Curious3141]

Nice job there Curious.

To find the probability of the card colours turning out a certain way, an easier approach would be to reverse the deck and deal two cards from the top. What is the probability that the two cards are of the same colour? Of diff. colours?
I haven't tried this ( too lazy ) but you ought to get the same answer shouldn't you ?

Yes, you are completely correct. The situation is completely reversible. Reversing the draw, the first card choice is immaterial. The prob of getting a similar card with only 25 of that sort left in the deck is 25/51. Prob of dissim is therefore 26/51, and everything works out the same, with much less fuss.

That's a much nicer job than mine.

 

[arunbg]

Ok I tried it and it worked beautifully.
Here's what I did.

To find the required probability, rather than dealing cards till the 51st and 52nd, we can deal the cards in the opposite sense to give the same probability as I mentioned in my earlier post.

Simply find the probability of getting two cards of the same colour and different colours separately(in two draws) , multiply with the probability that you get your guess is right and you have the answer. Mathematically, this comes out as
P = P(getting red and black)P(guessing right) +
P(getting both black or both red)P(guessing right)
= 26/51*1/2 + 25/51*1 = 38/51

which is the same answer Curious obtained. Pretty simple isn't it ?
Of course Curious' expression for the general probability is better when the cards for which you have to calculate the probability are more towards the middle.

Edit: Curious didn't see your post

 

[HallsofIvy]

Of course, this problem had nothing to do with "Bayesian probability". I'm wondering why blumfeld0 would talk about "Bayesian probability" when in a more recent thread he has claimed not to know how calculate the probability of drawing a red ball from an urn containg 4 red and 4 white balls!

 

[Curious3141]

Of course, this problem had nothing to do with "Bayesian probability". I'm wondering why blumfeld0 would talk about "Bayesian probability" when in a more recent thread he has claimed not to know how calculate the probability of drawing a red ball from an urn containg 4 red and 4 white balls!

"Bayesian" has sort of a nice esoteric ring to it. It's the same reason why I've been seeing threads entitled "Relativity Theory" that discuss nothing more abstruse than F = ma.

 

[blumfeld0]

Thanks for all the great replies
it was in the same chapter as bayesian probability so i figured it was bayesian probability