Almost 2019

It is October 2018 and it is near to 2019. Included here is a problem involving 2019. The one who can post the correct answer first will post another problem involving 2019. Our target is to get 2019 problems here. So, here is the first problem:

There are 2019 prisoners in a prison. The prison warden decides to play a game with them. He lines them up in a row all of them facing another prisoner. Prisoner N will face Prisoner N-1. He takes 2019 black hats and 2019 white hats. He randomly puts one on each prisoner’s head. (He flips a standard coin to determine what hat to put on a prisoner’s head.) Everyone can see the colour of the hat of the person with any number lower than them. He explains the rules of the game and tells the prisoners that each one of them has to guess the colour of their hat or say pass. If any of them will guess incorrectly, everyone spends 10 more years in prison. If someone guesses correctly, everyone goes free. If someone guesses right but another guesses wrong, they spend 10 more years in prison. If everyone says pass, nothing happens. What is the probability that they spend 10 more years in prison if they play optimally?

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Almost 2019

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