We were given this problem but I'm not sure I understood the term "face value". Does it mean cards with "faces" like Jack, Queen, King? or Does it pertain to the value a card holds like 4 of Spades has the same value with 4 of hearts? Maybe someone else had a similar problem before, please tell me what "face value" means in context of these types of problems.
Here's the original problem:
Suppose you have a well-shuffled deck of 52 cards. On the average, how many pairs of consecutive cards on the deck have the same face value? Estimate the probability that such a pair exists using Poisson distribution.
I'm also confused with what a well-shuffled deck means. Initially, I am thinking that it suggests that all cards have the same chances of being drawn.
Let $X_i$ be an indicator random variable that is equal to $1$ if the card following it has the same face value, and $0$ otherwise
Then $\Bbb P(X_i) = \frac3{51}$
Now the expectation of an indicator r.v. is just the probability of the event that it indicates, so $\Bbb E(Xi) = \frac 3{51}$
and by linearity of expectation, $\Bbb E(X) =\Bbb E(X_1) + \Bbb E(X_2) +... \Bbb E(X_{51}) = 51\cdot \frac3{51} = 3$