Probability of guessing the colors of a deck of cards correctly 10 years ago when I was about 15 I sat down with a deck of shuffled cards and tried to guess if the next card in the deck would be red or black. In sequence I guessed 36 cards correctly as red or black, at that point my older bother came in and took told me what I was doing was stupid and I stopped being able to guess correctly. I would love to know what the probability of guessing 36 cards correctly in sequence is? Thank you  
 A: If you're aiming to guess the first 36 cards right with as high a probability as possible, then the best you can do is to choose some sequence of guesses with 18 reds and 18 blacks, and all those sequences are equally likely. 
Each of them has probability 
$\frac{(26\times 25\times 24\times\dots\times9)^2}{52\times51\times50\times\dots\times17}$
$=\frac{26!\times26!\times16!}{52!\times8!\times8!}$
$\approx 2.595\times 10^{-11}$. 
So, the chances are about 1 in 40 billion. 
(Assuming the pack of cards is shuffled)
It's easier than guessing 36 coin-tosses correctly, which would have probability $2^{-36}$ which is about 1 in 70 billion. 
A: If you just guess red or black with a 50/50 chance, without taking into account that memory of the previous cards can make a guess more accurate, then the probability of guessing 36 cards in a row correctly is:
$$\left(\frac 12\right)^{36}$$
For comparison, if all 7.125 billion people in the world were to attempt this, the probability that anyone in the world would succeed is
$$\left(\frac 12\right)^{36} ~~\times~~ 7.125 \times 10^9 = 10.4\%$$
Which makes your story somewhat unbelievable.
