Which has the higher probability?
A) A fair coin is tossed 26 times. Write down an expression for the probability of seeing exactly 13 heads and 13 tails.
B) A pack of 52 cards (containing 26 red and 26 black cards) is shuffled, and then 26 cards are dealt. Write down an expression for the probability that exactly 13 red and 13 black cards are dealt.
I think the probability for A) is $ {26 \choose 13} \frac{1}{2^{26}}$ and for B) it is $\frac{{26 \choose 13}{26 \choose 13}}{52 \choose 26}$. I don't have much intuition for which one should be bigger (I was hoping to see a way to tell which is bigger intuitively rather than comparing the probabilities e.g. with Stirling's formula). I was thinking that an argument for why (B) is bigger could be as follows:
Consider the difference between tossing 13 heads in a row so far and having dealt 13 red cards in a row so far. If you have tossed 13 heads, the probabilities for the future tosses are not changed. However, if you've dealt 13 red cards, then there are 13 reds and 26 blacks left in the deck - so there is a "correcting force" pushing you back to a more even split of reds and blacks in the 26 that you draw.