A post on another site, and several posts in the past, have asked about optimizing betting for Color of Blackjack. COB involves different betting ramps at different depths using an unbalanced count to obtain much of the benefit of true counting without true counting. Blackjack, when played with perfect basic strategy, has a house edge of less than 1%. But not every strategy works when it comes to winning at blackjack. This post – the first in a series – looks at some of the good and bad strategies for winning at blackjack. Using Basic Strategy Is a Good Approach to Winning at Blackjack. The optimal bets for nine penetrations are displayed. 26 cards to 130 cards cut off by half-deck increments. We start with the dark green line — a very poor penetration of 130 cards (2.5 decks out of 6 cut off). Bets range from $5 to $120. The bets ramp up very quickly with the max bet placed at. The basic strategy chart gives you the correct play for every possible situation at the blackjack table. It will show you whether you should hit, stand, split, double, or surrender. You look across the top of the chart for the dealer’s up card, and along the left side for the hand that you hold. The optimal back-betting split strategy will depend on whether the game is single-deck or multiple-deck, whether the dealer hits or stands on soft 17, whether or not doubling after splitting is allowed, and whether or not the European no-hole-card rule is in effect (i.e., dealer blackjack takes all.

• Blackjack Betting Strategy 1 3 2 6
• Blackjack Betting Strategy Cards
• Blackjack Best Betting Strategy
• Blackjack Betting Guide

Each decision you make while playing blackjack has a positive or negative
impact on your potential profit or loss. Every situation you have while playing
has a single best way to play.

If you make the correct play it either makes you the most money in the long
run or loses the least amount of money in the long run. This is called basic
strategy.

Some hands lose money in the long run and some win in the long run, so your
job is to make the best possible play to maximize the wins and minimize the
losses.

When you read about the expected house edge in blackjack games based on the
rules the percentages are based on perfect basic strategy. If you don’t use the
proper strategy you give the house a higher edge against you. This can add an
extra one or two percent to the house edge, depending on how far from the proper
strategy you stray.

You can usually find blackjack games with rules that offer a house edge of
less than 1%, and can often find games with a house edge under a half percent.
If you don’t use basic strategy you can be playing with a house edge of 1.5% to
3% instead of a half percent.

This quickly adds up.

Blackjack Betting Strategy 1 3 2 6

If your average bet is $100 and you play 100 hands per hour and you give the
casino and extra 1% you lose an extra $100 per hour. By giving them an extra 2%
it’s $200 per hour.

If you want to play blackjack the first thing you need to do is decide to
always make the best possible play.

You probably realize that there are hundreds of hands when you combine all of
the possible hands you can have with the different up cards the dealer can have.

The good news is that by using a simple chart you can quickly find the best
play. And you can use a strategy chart or card at the table while playing in a
live casino or when playing online.

If you play in a live casino some of the other players may try to pressure
you if you take too long to make a decision. But it’s none of their business and
you can play any way you want as long as you don’t hold up the game too long.

If you’re worried about holding up the game sit in the middle of the table or
to the dealer’s right hand side so you have longer before you have to act. This
gives you more time to consult your strategy chart before being force to make a
decision.

You don’t even need to memorize the best plays.

But with a little effort and time you can quickly memorize the most common
plays and in time memorize all of the best plays.

The next section has a chart and the following section has the hand by hand
explanations. Then you’ll find a section explaining a few ways to start
memorizing all of the plays.

Basic Strategy Chart

This is the most universal basic strategy chart and can be used in all
blackjack games with any rule combinations. A few small adjustments can be made
against certain rules combinations, but it becomes complicated to memorize
multiple charts. We’ve included a short section later explaining these
adjustments for the dedicated player.

The dealer’s up card is listed across the top. Your cards are listed down the
first column to the left. Simply go down the left column until you find your
cards and then go across to the column with the dealer’s card to find the best
play.

The key for each play is as follows:

• Hit – Hit
• Stand – Stand
• DblH – Double
• DblS – Double if permitted, otherwise stand
• Split – Split
• SplitD – Split if double after split is permitted, otherwise hit
• SUR – Surrender if permitted, otherwise hit

Expand Shrink

Text Version and Vision Impaired

Many players use the chart listed above but a few players prefer to read the
correct plays. Having the correct plays written out also is valuable to people
who have vision problems and use software to hear what’s on the page.

We’ve listed the proper plays for each situation below in four sections.
Simply find the section that describes your hand and follow the instructions.

Hands with an Ace or Soft Hands

Any starting hand you hold with an ace is called a soft hand. A soft hand is
when you have an ace that can be used as a one or 11. In the following
combinations, if you have more than two cards, simply add the cards that aren’t
an ace.

This can happen when you start with an ace and a different side card and hit.
Starting with an ace two and receiving a four after hitting leaves you with ace
two four. Looking at the chart you look at the line for a soft 17. Using the
list below you look at the one for ace six because the two and four add up to
six.

• Ace ace – Always split a pair of aces. Split aces as many times as allowed.
• Ace two or soft 13 – You always hit with a soft 13 and double if allowed
  against a dealer five or six.
• Ace three or soft 14 – Always hit and double against a five or six if
  allowed.
• Ace four or soft 15 – You always hit with a soft 15 and double if allowed
  against a dealer four, five, or six.
• Ace five or soft 16 – Always hit and double against a four, five, and six if
  allowed.
• Ace six or soft 17 – Double against a three, four, five, and six if allowed
  and otherwise hit.
• Ace seven or soft 18 – Stand against a dealer two, seven, and eight, hit
  against a nine, 10, or ace, and double if allowed against three, four, five, or
  six.
• Ace eight or soft 19 – Stand at all times.
• Ace nine or soft 20 – Always stand.
• Ace 10 – Blackjack! Always stand.

Pairs

Whenever you have a pair for your first two cards you need to decide if you
should split.

• Two two – Split against a dealer four, five, six, and seven, hit against a
  dealer eight, nine, 10, and ace, and against a dealer two or three split if you
  can double after split and hit if double after split isn’t allowed.
• Three three – Play your hand exactly the same as two two. If double after
  split is allowed split against a dealer two or three, otherwise hit against
  these two cards. Hit against a dealer eight, nine, 10, and ace, and split
  against a dealer four, five, sis, and seven.
• Four four – When double after split is allowed split against a dealer five
  and six, otherwise hit. Hit against all other dealer cards.
• Five five – Never split fives. Hit against a dealer 10 or ace and double
  against all other dealer cards if allowed.
• Six six – Hit against a dealer seven, eight, nine, 10, and ace. Split against
  a dealer three, four, five, and six. Split against a dealer two if double after
  split is allowed, otherwise hit.
• Seven seven – Split against a dealer two, three, four, five, six, and seven.
  Hit against a dealer eight, nine, 10, and ace.
• Eight eight – Always split eights.
• Nine nine – Split against a dealer two, three, four, five, six, eight, and
  nine. Stand against a dealer seven, 10, and ace.
• Ten ten – Always stand with a total of 20.
• Ace ace – Aces should always be split. If you receive another ace on one of
  your split aces you should split again. Do this as many times as possible.

Hard Totals

A hard total is any hand that doesn’t have an ace where you can use it as a
one or 11. In any two card starting hand you can use an ace as either one or 11,
but after you hit one or more times you can reach a total where the ace can’t be
used as an 11 without busting. When this happens you hold a hard hand.

• Hard four – The only hard four is a pair of twos. This is covered in the two
  two line under pairs. If splitting isn’t allowed then always hit with a hard
  four.
• Hard five – Always hit with hard five.
• Hard six – Hit against all dealer cards with a hard six.
• Hard seven – Always hit with a hard seven.
• Hard eight – Hit against all dealer totals with hard eight.
• Hard nine – If double is allowed, double against a dealer three, four, five,
  or six, otherwise hit. Hit against all other dealer cards.
• Hard 10 – When double is allowed, double against a dealer two, three, four,
  five, six, seven, eight, and nine, otherwise hit. Hit against a dealer 10 and
  ace.
• Hard 11 – Double against any dealer car except an ace if doubling is
  permitted, otherwise hit. Hit against a dealer ace.
• Hard 12 – Stand against a dealer four, five, and six. Hit against all other
  dealer cards.
• Hard 13 – Stand against a dealer two, three, four, five, and six. Hit against
  a dealer seven, eight, nine, 10, and ace.
• Hard 14 – Stand against a dealer two, three, four, five, or six and hit
  against seven or higher.
• Hard 15 – Against a dealer two, three, four, five, and six you should stand.
  Against a dealer seven, eight, nine, 10, and ace you should hit.
• Hard 16 – Stand against a dealer total of two, three, four, five, and six.
  Hit against all other dealer totals.
• Hard 17 – Stand against all dealer hands.
• Hard 18 – Always stand with a hard 18.
• Hard 19 – Stand against all dealer hands.
• Hard 20 – Always stand with a hard 20.
• Hard 21 – Stand against all dealer totals.

Surrender

If surrender is allowed you should surrender with a hard 15 against a dealer
10 and with a hard 16 against a dealer nine, 10, or ace. If surrender isn’t
allowed then hit in these four situations.

Adjustments for Different Rules

The chart and instructions above are designed for a game where the dealer
stands on a soft 17. If you play in a game where the dealer hits on a soft 17
you can make the following adjustments.

As we mentioned above, it can be complicated to keep two different charts
straight in your mind so learning the first chart is best. However, there are
only six hands that change so it’s possible to learn and use the differences.

Split hands

Instead of splitting eight eight against a dealer ace,
surrender if allowed, otherwise split.

Soft hands

With a soft 18 against a dealer two and with a soft 19 against a
dealer six double if allowed, otherwise stand.

Hard hands

With a hard 11 against a dealer ace double if allowed, hit
otherwise. With a hard 15 against a dealer ace surrender if permitted, otherwise
hit. With a hard 17 against a dealer ace, surrender if permitted and otherwise
stand.

Insurance or Even Money

Notice that there’s not a single hand that recommends taking insurance or
even money. This is because taking insurance is always a losing proposition.

When the dealer has an ace showing you have the option of making another bet
equal to the size of your first bet that pays two to one if the dealer has a 10
hole card. If you take insurance and the dealer has a 21 you lose your first bet
but get double on your second so you break even.

But when the dealer doesn’t have 21 you lose the insurance bet and your first
bet plays out as normal.

On the surface this may seem like a good bet, but a simple look at the math
behind the bet shows why you should never take insurance.

When the dealer has an ace showing she has a blackjack any time a 10, jack,
queen, or king is her down card. This means four of the 13 possible cards pay
off on the insurance bet.

But this also means that the other nine cards, two through nine and ace, make
you lose the insurance bet. This is a ratio of nine to four, with nine losing
cards and four winning cards. But the bet only pays two to one.

The odds of nine to four is worse than two to one, so it’s a losing bet.

Forget your normal hand because it doesn’t have anything to do with the
insurance offer. Insurance is just a bet on what the dealer has as a down card.
And since it’s offered at bad odds you can now see that it should be avoided.

This is just another way the casinos try to be sneaky and build a bigger long
term edge against you.

How to Memorize Basic Strategy

The easiest way to start memorizing basic strategy is to use the chart
provided above to make every playing decision. As you play you’ll start
recognizing the correct play for most hands before checking your chart.

Sign up for a free account at one of our recommended online casinos and start
playing the free blackjack games. You can play over 100 hands per hour in most
cases so you can get a great deal of practice in a short amount of time.

Eventually you’ll only need to check your chart on the most difficult plays
and if you play long enough all of the plays will come from memory.

The other way is more difficult for most players, but if you have a strong
mind for memory you can simply start memorizing the chart.

A quick tip is to group similar hands together.

All of the hard hands totaling four, five, six, seven, and eight are played
the same way. On all of these hands you always hit. If you’re playing a game
where doubling isn’t allowed, you can add all of the hard totals of nine, 10 and
11 to this. As a side note, you should avoid games where doubling isn’t allowed
as they have a higher house edge.

Other easy to learn hands include:

• Always split aces and eights.
• Soft 13 and 14 are played the same.
• Soft 15 and 16 are played the same.
• Soft 17 only has one dealer card played differently than soft 15 and 16.
• Hard 13, 14, 15, and 16 are played the same unless surrender is allowed.
• Hard 10 and 11 are the same except for one dealer card.
• Two two and three three are played the same.
• All of the hard totals 17 and above are played the same.
• All of the soft totals of 19 and above are played the same.

The majority of hands are hard hands and the complete hard hand strategy is
fairly simple. Memorize it first and you’ll find that you won’t need to refer to
the chart often.

The next chart to memorize after you’ve mastered the hard hands is the soft
hands. Most of them are straightforward as well, with the main changes of
knowing when to double down.

Once you have the hard and soft hands memorized the splits will come easy to
finish out your mental chart. You only need to learn the split rules for twos,
threes, fours, sixes, sevens, and nines.

The main thing is to not be afraid that you can’t learn the chart and to get
started right away. Almost anyone can improve their results and reduce the house
edge with a small amount of effort and time.

Card Counting

If you learn how to count cards you can play with a small edge against the
house while playing blackjack. This page is about basic strategy so we aren’t
going to dig into the realm of card counting, but it does have quite a bit to do
with basic strategy.

Card counters learn perfect basic strategy before they start trying to learn
about counting. If you can’t put in the work to learn perfect strategy you have
no hope of being a successful card counter.

Once you master basic blackjack strategy then you should investigate card
counting to see if it’s something you might want to learn. Counting cards online
won’t get you an edge because the software shuffles the cards after each deal,
but if you play live it might be able to help you win or at least play a break
even game.

Conclusion

Learning and using blackjack basic strategy gives you the best chance to win.
It reduces the house edge as much as possible and helps you have more winning
playing sessions.

Most players don’t take the time to learn basic strategy, but if you use the
information above you can quickly start using the best play for every situation.
Use the tips in the how to memorize basic strategy section and you’ll be a
master in no time.

And don’t forget to use the chart provided as you’re learning. Once you get
used to it you can find the proper play in a second or two.

My last post was about some common mistakes when bettingor gambling, even with a basic understanding of probability. This post is going totalk about the other side: optimal betting strategies using some veryinteresting results from some very famous mathematicians in the 50s and 60s.I'll spend a bit of time introducing some new concepts (at least to me), setting up theproblem and digging into some of the math. We'll be looking at it from thelens of our simplest probability problem: the coin flip. A note: I will not becovering the part that shows you how to make a fortune -- that's an exercisebest left to the reader.

Background

History

There is an incredibly fascinating history surrounding themathematics of gambling and optimal betting strategies. The optimalbetting strategy, more commonly known as the Kelly Criterion, was developed in the 50s byJ. L. Kelly , a scientistworking at Bell Labs on data compression schemes at the time. In 1956, he madean ingenious connection between his colleague's (Shannon) work on information theory,gambling, and a television game show publishing his new findings in a papertitled A New Interpretation of Information Rate (whose original title wasInformation Theory and Gambling).

The paper remained unnoticed until the 1960s when an MIT student named Ed Thorptold Shannon about his card-counting scheme to beat blackjack. Kelly's paperwas referred to him, and Thorp started using it to amass a small fortune usingKelly's optimal betting strategy along with his card-counting system. Thorpand his colleagues later went on to use the Kelly Criterion in othervaried gambling applications such as horse racing, sports betting, and even thestock market. Thorp's hedge fund outperformed many of his peers and it wasthis success that made Wall Street take notice of the Kelly Criterion. There is agreat book called Fortune's Formula 1 that details the stories andadventures surrounding these brilliant minds.

Surely, Almost Surely

In probability theory, there are two terms that distinguish very similarconditions: 'sure' and 'almost sure'.If an event is sure, then it always happens. That is, it is not possible forany other outcome to occur. If an event is almost sure then it occurs withprobability 1. That is, theoretically there might be an outcome not belonging tothis event that can occur, but the probability is so small that it's smallerthan any fixed positive probability, and therefore must be 0. This is kind ofabstract, so let's take a look at an example (from Wikipedia).

Imagine we have a unit square where we're randomly throwing point-sized darts thatwill land inside the square with a uniform distribution. For the entire square(light blue), it's easy to see that it makes up the entire sample space, so we wouldsay that the dart will surely land within the unit square because there is noother possible outcome.

Further, the probability of landing in any given region is the ratio of itsarea to the ratio of the total unit square, simplifying to just the area of agiven region. For example, taking the top left corner (dark blue), whichis 0.5 units x 0.5 units, we could conclude that (P(text{dart lands indark blue region}) = (0.5)(0.5) = 0.25).

Now here's the interesting part, notice that there is a small red dot in theupper left corner. Imagine this is just a single point at the upper leftcorner on this unit square. What is the probability that the dart lands on thered dot? Since the red dot has an area of (0), (P(text{dart landson red dot}) = 0). So we could say that the dart almost surely does not landon the red dot. That is, theoretically it could, but the probability of doingso is (0). The same argument can be made for every point in the square.

The dart actually does land on a single point of the square though, so eventhough the probability of landing on that point is (0), it still doesoccur. For these situations, it's not sure that we won't hit that specificpoint but it's almost sure. A subtle difference but quite important one.

Optimal Betting 2

Optimal Betting with Coin Tossing

Imagine playing a game with an infinite wealthy opponent who will always takean even bet made on repeated independent tosses of a biased coin.Further, let the probability of winning be (p > frac{1}{2}) andlosing be (q = 1 - p)3, so we have a positive overall expected valuefor the game 4. You start with (X_0) of initialcapital. Question: How much should we bet each time?

This can be made a bit more concrete by putting some numbers to it.Let's say our coin lands on heads with a chance of (p=0.53),which means tails must be (q=1-p=0.47). Our initial bankroll is(X_0=$100,000). How much of this ($100,000) should we bet on the firstplay?

Let's formalize the problem using some mathematics. Denote our remaining capitalafter the k'th toss as (X_k) and on the k'th toss we can bet (0leq B_k leq X_{k-1}). Let's use a variable (T_k = 1) if thek'th trial is a win, and (T_k=-1) for a loss. Then for the n'th toss, we have:

One possible strategy we could use is to maximize the expected value of(X_n). Let's take a look at that:

Since (p - q > 0) this will have a positive expected payoff. To maximize(E(X_n)), we should maximize (E(B_k)) (this is the only variable wecan play with), which translates to betting our entire bankroll at each toss.For example, on the first toss bet (B_0 = X_0), on the second toss (if we wonthe first one) bet (B_1 = 2X_0) and so on. It doesn't take amathematician to know that is not a good strategy. Why? The probability ofruin is almost sure (ruin occurs when (X_k = 0) on the k'th toss).

If we're betting our entire bankroll, then we only need one loss to lose allour money. The probability of ruin is then (1 - p^n) for (n) tosses (everyoutcome except winning on every toss). Taking the limit as (n) approaches infinity:

So we can see that this aggressive strategy is almost surely 5 going to result in ruin.

Another strategy might be to try and minimize ruin. You can probably already intuitthat this strategy involves making the minimum bet. From Equation 2, this isnot desirable because it will also minimize our expected return. This suggests that wewant a strategy that is in between the minimum bet and betting everything (duh!).The result is the Kelly Criterion.

The Kelly Criterion

Since our maximum bet is limited by our current bankroll, it seems plausible thatthe optimal strategy will always bet relative to our current bankroll. Tosimplify the math, we assume that the money is infinitely divisible. However,it should be noted that this limitation doesn't really matter too much when ourcapital is relatively large compared to the minimum divisible unit (thinkmillions vs. cents).

If on every toss, we bet a fraction of our bankroll (known as 'fixed fraction'betting), (B_k = fX_{k-1}), where (0 leq f leq 1), we canderive an equation for our bankroll after (S) successes and (F) failuresin (S+F=n) trials:

Notice that we can't technically ever get to (0) but practically there is a minimumbet and if we go below it, we are basically ruined. We can just re-interpretruin in this manner. That is, ruin for a certain strategy is when we willalmost surely go below some small positive integer (epsilon) as thenumber of trials (n) grows i.e., (lim_{nrightarrow infty}P(X_nleq epsilon) = 1).

Now let's setup what we're trying to maximize.We saw that trying to maximize the expected return leads us to almost surelyruin. Instead, Kelly chose to maximize the expected exponential growth rate.Let's see what that means by first looking at the ratio of current bankroll toour starting bankroll:

So (G(f)) represents the exponent (base (e)) on how fast ourbankroll is growing. Substituting Equation 4 into (G(f)):

Now since (G(f)) is a random variable, we want to maximize the expectedvalue of it (which we denote as (g(f))):

The last line simplifies because the expected proportion of successes andfailures is just their probabilities 6. Now all we have to do is a simpleexercise in calculus to find the optimal value (f^*) that maximizes (g(f)):

So we now have our optimal betting criterion (for even bets), fractional betswith (f^*=p-q).

Another interesting behavior of varying our fractional bets can be gleaned bygraphing (G(f))7:

We can see that our (f^*) maximizes the growth rate. However, there is a point(f_c) where our growth rate becomes negative. This implies that if weover-bet (f > f_c), we will almost surely reach ruin (because we have anegative growth rate). The following (summarized) theorem from Thorp's paperstates this more precisely:

1. If (g(f) > 0), then (lim_{nrightarrow infty}X_n = infty) almost surely.

2. If (g(f) < 0), then (lim_{nrightarrow infty}X_n = 0) almost surely.

3. Given a strategy (Theta^*) and any other 'essentially different strategy' (Theta), we have (lim_{nrightarrow infty}frac{X_n(Theta^*)}{X_n(Theta)} = infty) almost surely.

From this theorem, we can see that if we pick a fraction such that (g(f)> 0), then we'll almost surely tend towards an increasing bankroll.Conversely, if we pick a fraction (g(f)<0), then we will almost surelyresult in ruin. This matches up with our intuition that over-betting iscounter-productive.

(Continued from Example 1)Suppose we have our even-bet coin toss game and the probability of heads is(p=0.53) and probability of tails is (q=0.47). Our initialbankroll is ($100,000) (big enough that the minimum bet isn't reallysignificant). Applying our optimal betting criteria, on our first playwe should bet (f=p-q=0.53-0.47=0.06) or (6%) of our bankroll, translating to($100,000 * 6% = $6,000). Assuming we win the first play, we should bet($106,000 * 6% = $6,360) and so on.

If we bet less than (6%), we will still be increasing our bankroll but not atthe optimal rate. We can also bet more than (6%) up to the theoretical point (f_c)such that (g(f_c)=0) with the same result.We can numerically determine this turning point, which in this case is(f_c approx 0.11973). So betting more than roughly 11.9% will almostsurely cause us ruin.

We can also compute the expected exponential growth rate using our optimal(f^*= 0.06):

So after (n) plays, a player can expect his bankroll to be(e^{0.001801n}) times larger. A doubling time can be computedby setting (e^{0.001801n}=2), resulting in (napprox 385) plays.

Betting with Uneven Payoffs and Other Variations

We've so far only looked at games with even payoffs. We can generalize this result.If for each unit wagered, you can win (b) units, we can derive a modified versionof Equation 7:

Solving for the optimum yields (f^*=frac{bp-q}{b}).

Another variation is when you can make multiple simultaneous bets such as whenmultiple players share a single bankroll. Going through a similar exercise, wecan derive values for (f_1^*, f_2^*, ldots) assuming the games playedare independent. When two players are playing the same game (e.g. same tablefor Blackjack), the bets are correlated and adjustments must be made.Additionally, we can analyze more complex situations such as continuous (ornearly continuous) outcomes like the stock market which require a more thoroughanalysis using more complex math. See Thorp's paper for more details.

Conclusion

Kelly's optimal betting criterion is an incredibly interesting mathematicalresult. However, perhaps what is more interesting is that this theoretical resultwas put into practice by some of the very mathematicians that worked on it!Thorp has had wild success applying it in various situations such assports betting, Blackjack and the stock market. Of course by itself thecriterion isn't much use, it is only once you've found a game that has apositive expected value that you can put it to use. I would go into how to dothat but I think I've written enough for one day and as I said, it's best leftas an exercise to the reader.

References and Further Reading

• The Kelly Criterion in Blackjack Sports Betting, and the Stock Market by Edward O. Thorp.

• Optimal Gambling Systems for Favorable Games, E. O. Thorp, Review of the International Statistical Institute Vol. 37, No. 3 (1969), pp. 273-293 .

• William Poundstone, Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street. 2005. ISBN 978-0809045990. See also a brief biography of Kelly on William Poundstone's web page.

Blackjack Betting Strategy Cards

William Poundstone, Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street. 2005. ISBN 978-0809045990. See also a brief biography of Kelly on William Poundstone's web page.

This whole section just basically summarizes (with a bit more step-by-step for the math) the paper 'The Kelly Criterion in Blackjack Sports Betting, and the Stock Market'. So if you're really interested, it's probably best to check it out directly.

It doesn't really matter if the bias is heads or tails. The point is that you get to pick the winning side!

The expected value of winning for bet (B) is (Bp-Bq = B(p-q) > 0) since (p > q).

Almost surely here because it's theoretically possible that you can keep winning forever but it's such a small possibility that it basically can't happen. This is analogous to the red dot in the unit square.

The expected value of a binomial distribution (e.g. coin tossing) is just (np). So (np/n = p).

Blackjack Best Betting Strategy

Blackjack Betting Guide

Image from 'The Kelly Criterion in Blackjack Sports Betting, and the Stock Market'.