# Probability that two drawn cards values are the same after an interchange

Shuffle a full deck of cards and divide it into two stacks of 2626
cards. A card is taken from the top of the first stack, and, after
its value is observed, put into the second stack. The second stack is
then reshuffled, a card is dealt from the top, and its value observed.

What is the probability that the two values are the same?

The solution in my text is a follows:

Let II denote the event that the interchanged card is selected.
Let SS be the event that the two values are the same.
Then
P(S)=P(I)P(S|I)+P(Ic)P(S|Ic)=127⋅1+2627351.P(S)=P(I)P(S|I)+P(I^c)P(S|I^c)=\frac{1}{27}\cdot 1+\frac{26}{27}\frac{3}{51}.

I came to the same conclusion in my own work, except for the 3/513/51.

Why is P(S|Ic)=3/51P(S|I^c)=3/51 as opposed to 1/511/51?

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When you say “value” do you mean, say, that drawing 3♣3\clubsuit and then 3♢3\diamondsuit is a success?
– lulu
Oct 20 at 19:56

Yes, value here refers to the rank of the cards.
– Zermelo’s_Choice
Oct 20 at 19:57

2

Well, then there are 351\frac 3{51} choices that match value without being the same card. How do you get 151\frac 1{51}?
– lulu
Oct 20 at 19:58

I see my mistake now. For some reason I was assuming there was only one card with matching rank.
– Zermelo’s_Choice
Oct 20 at 20:01

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