Question: We shuffle a deck of 52 cards and then flip them one by one. Let X=Number of times we see 3 number cards in a row. The numbered cards are (2,...,10). Find E(X).
This question is from one of my recommended exercises. Not for homework, or marks.
I am not entirely sure how to solve this problem. I know that the number of numbered cards in a deck is 36. therefore, the probability of finding 3 in a row would be $P=(\frac{36}{52} \cdot \frac{35}{51} \cdot \frac{34}{50})$. However, I am not sure where to go from here.
The solution is $\frac{210}{13}$.
I have seen some solutions online where you multiply P by 52, i.e: $P(X)=P \cdot 52$, which yields a similar answer to the back of my text book.
I would really like to understand this problem.
Thank you in advance.
The right answer is $50\times P$ where your $P=(36\times35\times34)/(52\times51\times50)$ is the chance of getting three number cards on three successive draws. Where does the $50$ come from? There are that many opportunities for seeing three number cards in a row, namely with draws $d,d+1,d+2$ for $d=1,\ldots,50$. The expected number of successes is the number of trials (oportunities) times the chance $P$ that any given trial will succeed.