Hypothetical scenario: The game is No Limit Hold'em. You're playing a friendly game with two friends, and for fun, all three of you decide to go all in before the flop for $100 each. For further amusement, your friends let you pick any two cards from the deck, with one caveat: You must pay a premium.

Q: How much would you pay for pocket aces?

A: Good question.

Here's the thing: Pocket aces (AA) is the best possible hand you can have pre-flop in Hold'em. AA is a favorite to win heads up against any other hand, and will win more than its fair share (i.e., AA will win more than 33% of the money) against any two hands. However, its value changes somewhat drastically depending on the other players' holdings. Moreover, how much we would be willing to pay is not the same as what we think the hand is worth.

That said, how much we would pay is dependent on our approximation of fair value, and the first thing we must do is lay the basic groundwork for proper valuation.

The fair value range
Below, we have six hand matchups that at least closely represent the full range of possible values for AA. In all six hands, we hold the AsAc (ace of spades, ace of clubs), and the other two hands represent our opponents' hands. Without getting too much into the details, Hand No. 1 closely represents the best-case match-up for AA, while Hand No. 6 at least closely represents the worst-case scenario. The "Win %" and "Tie %" figures for each matchup were taken from the odds calculator at CardPlayer.com.

Ours

Opp. 1

Opp. 2

Win %

Tie %

Value

Hand No. 1

AsAc

Ah7d

Ad7h

93.19%

1.48%

$181.05

Hand No. 2

AsAc

AhKh

AdKd

79.74%

1.14%

$140.36

Hand No. 3

AsAc

AhKh

QdQc

70.81%

1.07%

$113.87

Hand No. 4

AsAc

KhKd

QsQc

66.37%

0.40%

$99.51

Hand No. 5

AsAc

JhTh

6d5d

57.66%

0.19%

$73.17

Hand No. 6

AsAc

AhAd

Ts9s

1.77%

71.04%

$11.75

*Tie % includes both two-way and three-way splits

Basically, what the table says is that against virtually any two conceivable hands, AA is worth a premium -- the value net of the $100 we already put in the pot -- of between $11.75 and $181.05. The value of a given hand is derived by multiplying the "Win %" by $300 (the amount that is in the pot), multiplying the "Tie %" by $150 for two-way splits, multiplying the "Tie %" by $100 for three-way splits, and adding up the results. Take the result and subtract the $100 initial investment, and we have the net value shown in the last column.

Note that the two-way and three-way split breakdowns are not included in the table, but are accounted for in the "Value" result.

The result is an initial fair value range of $11.75 to $181.05. But given the knowledge we have about our subject (Hold'em poker), we can narrow this down even further.

Estimating fair value
One way we can estimate fair value is to assign probabilities to the various outcomes. For example, if Hand No. 1 occurs 1% of the time that we hold AA all in against two other players, its probability is 0.01. The sum of all possible outcomes is equal to 1. Then you would multiply the probability of each outcome by its expected net value and add up the values.

Recognize that Hands Nos. 1 (against two players with A7-offsuit) and 6 (against another player with AA) represent fairly extreme examples. Hand No. 2 -- pitting our AA against two players with AK-suited -- is a bit more realistic, although still fairly unlikely. Hand No. 5 (against Jack-10 suited and Six-Five suited) is another oddball hand and a rarity, but fairly well represents the worst-case scenario for valuation purposes. Meanwhile, Hands Nos. 3 (against AK-suited and QQ) and 4 (against KK and QQ) most closely represent the real-world hands where three players would get all the money in before the flop and thus receive heavy weightings.

In the example below, I have assigned a probability of 0.01 to Hand No. 1, 0.13 to Hand No. 2, 0.50 to Hand No. 3, 0.30 to Hand No. 4, 0.05 to Hand No. 5, and 0.01 to Hand No. 6. The result is a weighted expected value of $110.64.

Value

Probability

Weighted Value

Hand No. 1

$181.05

0.01

$1.81

Hand No. 2

$140.36

0.13

$18.25

Hand No. 3

$113.87

0.50

$56.94

Hand No. 4

$99.51

0.30

$29.86

Hand No. 5

$73.17

0.05

$3.66

Hand No. 6

$11.75

0.01

$0.12

Total

1.00

$110.64

But there's another way to look at it. My preference is to ignore Hands Nos. 1, 2, and 6 entirely, as 1 and 6 are extreme, and 2 overstates the value of AA, in my opinion. Common sense might say that fair value lies somewhere between Hands Nos. 3 and 4, or between $99.51 and $113.87. In other words, AA is probably worth around $99.51 to $113.87, and is certainly worth more than $73.17 (the value of Hand No. 5).

Google as an example
Naturally, stocks can be evaluated the same way. Take Google (NASDAQ:GOOG) for example, which is currently trading at about $380 per share. Suppose your analysis indicates that, given potential outcomes, the stock has a 5% chance of being worth $450 today, a 13% chance of being worth $400, a 30% chance of being worth $350, etc. You would plug in the range of values, multiply them by their probabilities, and then add up the values. In our example, Google is worth $323.50 per share. Note that the figures in the table below are for illustrative purposes only.

Value

Probability

Weighted Value

Scenario No. 1

$450

0.05

$22.50

Scenario No. 2

$400

0.13

$52.00

Scenario No. 3

$350

0.30

$105.00

Scenario No. 4

$300

0.30

$90.00

Scenario No. 5

$250

0.20

$50.00

Scenario No. 6

$200

0.02

$4.00

Total

1.00

$323.50

You can see the importance of taking a company you're looking at and truly understanding it. A thorough understanding gives you the tools you need to properly value the stock.

Searching for value
Note that it's far more important that we get value than it is that we precisely value the stock. By that token, if you thought Google was worth $300 to $350 per share, you definitely wouldn't pay more than $350 for a share, and you should probably avoid paying even $300.

I should also point out that AA is a premium hand -- a blue chip, a best in class. It's the Hold'em equivalent of Berkshire Hathaway. As such, you might not require as big of a discount as you would for a lesser-quality hand. Thus, where pure value investors might say that the right to play AA is probably worth about $99.51 to $113.87 per hand, and they might normally require a 40% discount to play (i.e., they would pay $60 to $70 or so for the hand), they'd probably loosen up their requirement somewhat here.

Personally, I tend to look at things a little differently. When I look at a top-notch company that I would love to own, I'll have a general but conservative range of fair values. And usually, I'll buy the stock when it hits a price that I'm confident can't be right.

Riverboat casino operator and Motley Fool Hidden Gems selection Ameristar Casinos (NASDAQ:ASCA) is a perfect example: Over the past three years, I've followed the company. There have been a number of times when I've looked at the stock after a big drop and said that the company may or may not be worth more than 7.0 to 8.0 times EBITDA, but it's certainly worth more than 5.5 to 6.0 times EBITDA (here's one). I basically looked at slot makers International Game Technology (NYSE:IGT) and WMS Industries (NYSE:WMS) the same way (see IGT: Time to Buy? and The New WMS Industries).

So how much would you pay for AA?
To answer our initial question: AA is probably worth around $99.51 to $113.87 in this scenario, and is certainly worth more than $73.17. That said, I'd probably be willing to pay some discount to $99.51 for AA, and I'd load the boat at prices approaching $73.17 (our worst-case scenario). At the same time, I'd be mostly indifferent to the proposition at $99.51 to $113.87, and look for better bets elsewhere at a price above $113.87.

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Fool contributor Jeff Hwang owns shares of Berkshire Hathaway, Ameristar Casinos, International Game Technology, and WMS Industries. The Motley Fool has an ironclad disclosure policy.