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A standard deck of cards contains $52$ cards.  These $52$ cards are arranged in a row, at random.  Find the expected number of pairs of adjacent cards that are both Aces.

 Aug 22, 2023
 #1
avatar-1 
-1

To find the expected number of pairs of adjacent cards that are both Aces, we can consider the probability of each pair being Aces and then sum up these probabilities to calculate the expected value.

There are 52 cards in a standard deck. When these cards are arranged randomly, there are 51 pairs of adjacent cards. Each pair has the same probability of being Aces, which is 
4
52
×
3
51
=
1
13
×
1
17
52
4

 × 
51
3

 = 
13
1

 × 
17
1

 , since there are 4 Aces in the deck and 51 possible pairs after placing the first Ace.

So, the expected number of pairs of adjacent cards that are both Aces is:

Expected number
=
Number of pairs
×
Probability of pair being Aces
=
51
×
1
13
×
1
17
Expected number=Number of pairs×Probability of pair being Aces=51× 
13
1

 × 
17
1

 
Simplifying this:

Expected number
=
51
13
×
17
=
3
221

0.0136
Expected number= 
13×17
51

 = 
221
3

 ≈0.0136
Therefore, the expected number of pairs of adjacent cards that are both Aces is approximately Myjdfaccount
0.0136
0.0136.

 

If you face any difficulty please tell me .

Best Regards,

[ DR. Ginstioniff ]

 Aug 22, 2023
 #2
avatar+183 
-1

Hello "Dr." Ginstioniff. May I ask how you got your PhD? Was it on posting incoherent ChatGPT spam? 

plaintainmountain  Aug 22, 2023
edited by plaintainmountain  Aug 22, 2023

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