# I was born on Wednesday

Probability is a tricky thing. There are a lot of nonsensical answers to be had. I just read an article about the recent Gathering for Gardner meeting that took place. Gathering for Gardner is a unique meeting for mathematicians, magicians and puzzle makers where they get together and talk about interesting things. The meetings were inspired by Martin Gardner, one of the awesomest dudes of our time, who unfortunately just passed away. The question put to the floor was the following:

"I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"

Think about that for a moment. Not too hard though. The answer turns out to be surprising. Upon reading the question, I thought about it for a long time and managed to confused myself entirely. Thinking I had gone crazy, I wrote a little python script to test the riddle, which only left me more convinced I had gone insane. I've spent most of the night thinking about it, and after making it half way to crazy, I've come around and am momentarily convinced the puzzle makes perfect sense. I'm going to attempt to convince you it makes perfect sense, but I plan on doing it in steps so as to reduce the bewilderment.

#### Playing Cards

Forget the question. Lets play a game of cards. You shuffle a deck and deal me two cards:

I accidentally flip one of them over.

Whats the probability that my other card is red? Well, that ones easy, its about half. Sure, its not exactly a half, knowing that the deck is finite and that the draws are done without replacement, knowing that the card showing is a red one means that there are only 52/2 - 1= 26 -1 = 25 red cards out of a deck of 52-1 = 51 cards giving a probability of 49%. But its basically a half. Lets do over, deal me two cards:

Darn, I flipped one of them over again:

Whats the probability that my other card is red? About a half still. (Sure, this time its really 26/51 = 51%). Nothing mysterious going on. Do over again. Deal me two cards:

This time I'll ask a little trickier question. Whats the probability that both my cards are red? Ah, well its about 1/2 * 1/2 or about 1/4 = 25%. (The real answer is 24.5%) Alright smarty pants. Whats the probability that I had a red card and a black one? Well, that ought to be about 1/2 (Real answer 51%). All in all, I could have a red card, then a black one (RB), or a black one, then a red one (BR), or a red one then a red one (RR) or a black one then a black one (BB). 4 distinct possibilities, each of which are equally likely, so the above two answers make complete sense. There is only one way in four to get both red cards, but two ways out of four to have both a red and a black. So far so good. Lets ask a different question. Now I'm going to get a bit obtuse. You deal me two cards. Now you ask me.

Hey Alemi, do you have a red card?

Meaning, do I have at least one red card. I respond, "Yes." Now, go with your gut. You know I have at least one red card. What do you reckon the color of the other one is? Probably black you say? You'd be correct. Looking at our breakdown above, I could have gotten RR, RB, BR, or BB as my cards dealt. Each was equally likely, but now you know something else. You know that I have at least one red card, so we only have three possibilities left, I either have RR, RB, or BR. Each of which was equally likely. So whats the probability that my other card is black? About 2/3 or 67%. (Real answer: 67.5%) Alright, same situation. You deal me two cards, I reveal that I have at least one red one. Whats the probability that my other card is red? Well, obviously 1/3 or 33% (Actually 32.5%) since this is the opposite question to the one directly above, and follows from the same reasoning. Fine. No problems. All of this makes sense.