Expected Value for consecutive cards

Question

If I have a deck of cards and $Z$ is the no. of consecutive pairs of cards in our deck which are both black, how can I find the expectation of $Z$?

Answer 1

Let $(a,b)$ be a pair of cards in your deck. The probability that $(a,b)$ is a black-black pair is $25/102$: as @lulu pointed it out $$\mathbb P((a,b) \ \textrm{black})=\mathbb P(a \ \textrm{black}) \times \mathbb P(b \ \textrm{black} | a \ \textrm{black})=(1/2)\times (25/51)=25/102.$$

Let $X_{(a,b)}$ be the random variable defined as $1$ if $(a,b)$ is a black pair and $0$ otherwise.

Then $Z=\sum_{pairs (a,b)} X_{(a,b)}$.

By linearity of expectation you have $\mathbb E[Z]=\sum_{pairs (a,b)}\mathbb E [X_{(a,b)}] = 51\times(25/102)=25/2=12.5$ because there are 51 consecutive pairs.