## Poll

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**43 members have voted**

Quote:GialmereIt's Toughie Tuesday. Let's waste time playing...

While killing some time at your desk one afternoon, you fire up a new game of Solitaire on your computer (specifically the version where you deal out three cards from the deck at a time). But your boredom quickly turns to rage because your game is unplayable — there's nothing you can do on the board and you can flip through your deck, but you never have any legal moves!

Rounded to the nearest quarter percent, what is the probability of being dealt such a nightmare scenario?In America, Klondike is so popular that if you simply say you're playing Solitaire it's assumed that's the game you're playing. This is somewhat baffling since Klondike has a very low win ratio compared to most other solitaire card games. What are the odds that you'll win? Well, according to Wikipedia...Quote:WikipediaThe probability of being able to win a game of Klondike with best-possible play is not known, although Hoyle's Rules of Games suggests the chances of winning as being 1 in 30 games. The inability of theoreticians to precisely calculate these odds has been referred to by mathematician Persi Diaconis as "one of the embarrassments of applied probability".

The game enjoyed a surge in popularity when it was included as part of the Windows operating package. The idea was to get users comfortable using a mouse by dragging and dropping red 6s onto black 7s. The result was workers wasting endless hours playing the game. But hey, it was worth all that losing to see those cards fly around the screen when you finally won right? According to Microsoft, Solitaire was its most popular program for many years.

Klondike was also called Canfield in America, perhaps because it was once a casino game at the Canfield Casino in Saratoga Springs, New York. I've heard of Solitaire in casinos but have never seen it myself. I do think that the reason "Las Vegas Scoring" is so popular on computer versions is that you can lose the game but still claim a minor victory if you got enough cards up top to end with a positive cash amount.

Anyways, although we may never know the true odds of winning Klondike, the above puzzle (from the Riddler) is solvable (but not a cakewalk).

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Let’s lay out the things we need to consider to tackle this problem. The first thing is the board of cards we’ve arranged on the table to begin the game. That board includes seven face-up cards (and 21 face-down cards). The second thing is the deck that we’ll deal from during the game. That deck includes the 24 remaining cards, but because we’re dealing three cards at a time, we only care about eight of them (the ones that will appear at the top of the three-card draws). These two sets — the seven face-up cards and eight in the deck — will determine whether we’ll play a nightmare game of Solitaire in which we have no legal moves.

What’s left to do is an intricate counting problem involving some very big numbers. In what follows, I’ve adapted the approach of solver Jacob Kes, who was kind enough to provide the code he used for counting card combinations.

Brute force is always the most elegant solution! The key criteria in this problem are a card’s rank (that is, whether it’s a number or a face card) and its color (red or black). You can use a computer script to find all the possible combinations for the seven face-up cards by rank and color only — for example, a red five, a black seven, a red jack, etc.3 But we can’t forget the suits!

For each set of seven number-and-color combinations, there’s a corresponding number M of possible combinations of seven cards (now taking suit into account). For each of these combinations, you can calculate how many of the remaining cards (of which there are 45 and might end up in the deck) wouldn’t allow us any legal Solitaire moves — let’s call this number N. That’s how many are neither aces, which can be moved to a designated area on the Solitaire table, nor are cards that can be moved onto a face-up card.

This means that, for a given seven cards, our chance of a nightmare deal is of P = (N choose 8)/(45 choose 8). Each face-up card combination has a 1/(52 choose 7) chance of occurring, so for a total probability, you just need to sum M*P/(52 choose 7) over all of the face-up number-and-color combinations. This winds up being 643,746,385,468/257,479,369,193,475, or about 0.0025, or about 1/400.

In addition to walking through the intricate math to arrive at this number, Laurent Lessard calculated the probability of a nightmare game in the version where you deal one card at a time. It’s significantly lower: about 1.8⋅10^−7

Good luck avoiding a nightmare game, you brave, solitary warriors.

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Henry and Tom decide to bet on a coin flip. Henry wins on heads, Tom wins on tails.

It’s $1 per flip and they are really bored, so they decide to do one million flips. After each flip, the loser pays the winner $1. The players may bring any bankroll they want, but if a player goes bankrupt he automatically loses the game. Tom is very wealthy and tells Henry he's bringing a million dollars to the game. Henry is not rich and decides to bring an amount that will give him 50% confidence of not going bankrupt. How much should Henry bring?

I get the same answer*** as the Wizard above

***: $337 when rounded

Formula used:

~0.67449σ = "50% of proportion within"

n = "number of coin flips" = 1 million

~0.67449 x0.5 x (n^0.5) = 0.337245 x (1,000,000^0.5) = 0.337245 x 1000 = 337.245

Note: I don't use this (or a variation of this formula) very often, so I could have used it incorrectly / applied it wrong.

Quote:Ace2I respectfully disagree and suggest that you double (cough, cough) check your answers

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Because I am a relative beginner at this, can you give me a hint? (in spoilers if you like)

Something like, "the formula you used was correct in this situation, but the numbers are wrong", thanks

Quote:Ace2This hint was in my last reply!

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Thanks.

But when I looked up the SD for a single coin flip online, I found it was 0.5 from multiple sources (the above figure assumes 1 for the SD).

If 674 is closer to correct, please explain why my original POSTED answer was wrong.

Edited (about 455pm, Pac time): In the spoiler see word in capitals, for the edit.