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"A deck of cards is shuffled and then divided into two halves of 26 cards each. A card is drawn from one of the halves, it turns out to be an ace. The ace is then placed in the second half-deck. The half is then shuffled and a card is drawn from it. Compute the probability that this drawn card is an ace."

Source : A First Course in Probability, Sheldon Ross, Chapter 3, Exercise 37

(My intention was not to be lazy and let the community do all the work for me, but I think that is what has been misunderstood seeing the 2 dislikes this question received. I just thought I should not clutter the question with more text than necessary)

What I've tried : Probability that the second half already contained 0,1,2 or 3 aces before the ace from the first half was added to it. Then, when the new ace was added to it from the first half, we could calculate the probabilities of drawing an ace considering each of these cases, and add them to get the answer. But I am not sure how do I calculate the P(second half contained 0/1/2/3 aces)..

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  • $\begingroup$ @Brad: I know Bayes' theorem, but I'm not sure how it is to be applied in this case. $\endgroup$ – user118102114 Jan 6 '12 at 6:13
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This is a nice application of Bayes' theorem/conditional probabilities. Well, not really Bayes' theorem since we aren't inverting any conditional probabilities, but we need to express a probability as a weighted sum of conditional probabilities.

Let $A$ be the event that we draw an ace the second time, and let $P(i)$ be the probability that there are $i$ aces originally in the second pile and $P(A|i)$ be the probability that we draw an ace conditional on the second pile starting with $i$ aces. We have

$$P(A)=\sum_{i=0}^{3} P(A|i)P(i)$$

It is easy to compute $P(A|i)=\frac{i+1}{27}$. Computing $P(i)$ is slightly more involved. We need to compute the number of possible half-decks with $i$ aces, which we will denote $N_i$, and then $P(i)=\frac{N_i}{\sum_{j=0}^{3}N_j}$.

If we have $i$ aces and $26-i$ other cards (out of the 48 non-aces), then there are $\binom{4}{i}\binom{48}{26-i}$ possible hands.

Computing the answer is now a matter of plugging into the formula. Note that true blue anil's answer is slightly incorrect (if you are trying to do a comparison between the responses), as when we computed our probabilities, we divided by $N_0+N_1+N_2+N_3$, and not by $\binom{52}{26}=N_0+N_1+N_2+N_3+N_4$. Also note that if one wants an exact numerical answer, the binomial coefficients can be expanded in terms of factorials, and after getting a common denominator, there will be significant cancellation.

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Here are two solutions:
Let $A$ be the event an ace is drawn from the second half
Let $I$ be the event that the interchanged card is selected

$P(A) = P(A|I)\cdot P(I) + P(A|I^c)\cdot P(I^c)$
$P(A) = 1\cdot\dfrac{1}{27} + \dfrac{3}{51}\cdot\dfrac{26}{27}$
$P(A) = \dfrac{43}{459} = 0.094$

Note that $P(A|I^c) = \dfrac{3}{51}$ because there are 3 aces to choose from and all 51 other cards are equally likely to be in the second half

Second Solution:
Let $A$ be the event an ace is drawn from the second half
Let $N_i$ be the event that $i$ aces are in the second half with $0 \leq i \leq 3$

$P(A) = \sum_{i=0}^3 P(A|N_i)\cdot P(N_i)$
$P(A) = P(A|N_0)\cdot P(N_0) + P(A|N_1)\cdot P(N_1) + P(A|N_2)\cdot P(N_2) + P(A|N_3)\cdot P(N_3)$ $P(A) = \dfrac{1}{27}\cdot P(N_0) + \dfrac{2}{27}\cdot P(N_1) + \dfrac{3}{27}\cdot P(N_2) + \dfrac{4}{27}\cdot P(N_3)$

Note that there are only 3 aces to choose from and 51 cards total, since one ace is in the first half

$P(N_0) = \dfrac{\dbinom{3}{0}\dbinom{48}{26}}{\dbinom{51}{26}}$
$P(N_1) = \dfrac{\dbinom{3}{1}\dbinom{48}{25}}{\dbinom{51}{26}}$
$P(N_2) = \dfrac{\dbinom{3}{2}\dbinom{48}{24}}{\dbinom{51}{26}}$
$P(N_3) = \dfrac{\dbinom{3}{3}\dbinom{48}{23}}{\dbinom{51}{26}}$

$P(A) = \dfrac{43}{459} = 0.094$

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    $\begingroup$ I have downvoted this answer, as (1) the lack of appropriate formatting using MathJax makes it very hard to read and (2) the question already has several answers, and I don't see how this answer adds much that hasn't already been said. $\endgroup$ – Xander Henderson Jun 9 '18 at 23:18
  • $\begingroup$ All the other answers are incorrect. The probability is 0.094, not 0.10678 $\endgroup$ – nimo956 Jun 9 '18 at 23:29
  • $\begingroup$ I cleaned up the formatting. $\endgroup$ – nimo956 Jun 9 '18 at 23:47
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# of ways the 2nd half-deck could have 0,1,2,or 3 aces $=$

$$\binom{48}{26}+\binom41\binom{48}{25}+\binom42\binom{48}{24}+\binom43\binom{48}{23}$$

It now has 1 ace added to it, so

$$\begin{align*} \mathrm{P}[\text{ace drawn from it}] &= \frac1{27}\binom{52}{26}^{-1}\left(\binom{48}{26}+8\binom{48}{25}+18\binom{48}{24}+16\binom{48}{23}\right)\\\\ &\approx .10088 \end{align*}$$


Erratum

As pointed out by Byron Scott, ${\dbinom{52}{26}}^{-1}$ should be changed to ${\left(\dbinom{52}{26} - \dbinom{48}{26}\right)}^{-1}$ to get the correct sample space, which changes the answer to $\approx 0.10678$

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  • $\begingroup$ Since the original first half contained an ace, not all $\binom{52}{26}$ splits are actually possible. $\endgroup$ – Brian M. Scott Jan 6 '12 at 9:51
  • $\begingroup$ Is this is really the right answer? statweb.stanford.edu/~serban/116/sm1.pdf Even my answer is also 0.10088 but this pdf puts me in doubt or is it just rounding $\endgroup$ – Ketan Jan 25 '18 at 19:30
  • $\begingroup$ @ketan: I have added an erratum to my answer. As for the site you mention, the two parts actually give different answers ! $\endgroup$ – true blue anil Jan 26 '18 at 13:14
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Let $A$ be the event that an ace is drawn from the first half of the deck.

Let $X$ be the count of aces in the second half-deck before the additional ace is added from the first deck.

$$\mathsf P(X=x\mid A) ~=~ \dfrac{\dbinom{4}{x}\dbinom{48}{26-x}\mathbf 1_{x\in\{0,1,2,3\}}}{\dbinom{52}{26}-\dbinom{48}{26}}$$

Let $B$ be the event of drawing an ace after adding one and shuffling.   There are now $X+1$ aces among $27$ cards.

$$\mathsf P(B\mid X=x, A)~=~\dfrac{x+1}{27}$$

By the Law of Total Probability.

$$\begin{align}\mathsf P(B\mid A)~=~& \sum_{x=0}^3 \mathsf P(B\mid X=x, A)\mathsf P(X=x\mid A) \\[1ex] ~=~& \dfrac{1\cdot \dbinom{3}{0}\dbinom{48}{25}+2\cdot \dbinom{3}{1}\dbinom{48}{24}+3\cdot\dbinom{3}{2}\dbinom{48}{24}+4\cdot \dbinom{3}{3}\dbinom{48}{23}}{27~\left(\dbinom{52}{26}-\dbinom{48}{26}\right)}\end{align}$$

Remark: Since we are given that event $A$ has happened, the probability that we seek is indeed conditional on this event: $\mathsf P(B\mid A)$

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