What is the expected number of runs of same color in a standard deck of cards?

Standard deck has 52 cards, 26 Red and 26 Black. A run is a maximum contiguous block of cards, which has the same color.

Eg.

• (R,B,R,B,...,R,B) has 52 runs.
• (R,R,R,...,R,B,B,B,...,B) has 2 runs.

What is the expected number of runs in a shuffled deck of cards?

• Apparently this has an easy solution. Commented Aug 19, 2010 at 1:38
• Want to post it ? Commented Aug 19, 2010 at 2:28
• why has this question/answer pair been imported in toto from the math SE? Commented Sep 20, 2010 at 21:28

Suppose $$X_n$$ denotes the color of the $$n$$th card in the shuffled deck.

Then note that the last card always denotes the end of a run. Other ends of runs are characterized by $$X_n\ne X_{n+1}$$ which indicates a run ending at $$n$$.

Note that $$P(X_n\ne X_{n+1})=26/51$$ (since once you fix a card, you can choose another card from remaining 51 out of which 26 will have a different color).

So summing up the indicators $$X_n\ne X_{n+1}$$ we get the number of runs -

$$\#\text{runs}=1+\sum_{n=1}^{51}\mathbb{I}_{X_n\ne X_{n+1}}.$$

So $$E(\#\text{runs})=1+\sum_{n=1}^{51}P(X_n\ne X_{n+1})=1+\sum_{n=1}^{51}26/51=27.$$

Source

• I am not convinced -- is P(X_n != X_(n+1)) independent from n? Commented Aug 19, 2010 at 21:36
• You ought to credit the person who really answered this question (George Lowther), especially because you copied his answer with only trivial changes. See math.stackexchange.com/questions/2763/… .
– whuber
Commented Aug 21, 2010 at 17:17
• @whuber Yes definitely, it's not my credit. I merely reproduced it here. Thanks for putting up that note. @Karsten It is shown by a counting argument - once you have fixed $X_n$ in 52 ways, there are 26 ways you can select non-matching $X_{n+1}$. So $P(X_n\ne X_{n+1})=\frac{52*26}{52*51}$. Commented Aug 22, 2010 at 12:35
• @Karsten If you are asking if the events $X_n \ne X_{n+1}$ are independent of $n$, they are not. But that doesn't matter since expectation of sum = sum of expectations even if the quantities are dependent. Commented Aug 22, 2010 at 12:38
• See also en.wikipedia.org/wiki/Wald–Wolfowitz_runs_test , where the mean number of runs matches that given here. Commented Sep 20, 2010 at 4:51