Our resident blackjack gurus explain EV, a fundamental gaming concept. Calculating the value of that single ten card against the dealer's 6 isÂ ...

This post is for people interested in how that calculation is done. Most people would try to calculate EV's in blackjack by simulations since it isÂ ...

Blackjack 101: Thinking in Terms of EV

Introduction Analysis of casino blackjack has been a recurring. been on accurate and efficient calculation of exact expected value (EV) of theÂ ...I often use the term expected value (or EV) when I'm writing about the. The math used to calculate EV is not very complicated, and I'll show inÂ ...

Poker is all about making the best decisions you can with limited information. In other words, +EV (plus expected value) decisions. Here's howÂ ...

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## Roulette Expected Value - How to Calculate Expected Values in Roulette How to calculate expected value blackjack

Using such a strategy, can we change the expected value of losses / gain. Let us try to calculate the probability of the dealer getting burst.This post is for people interested in how that calculation is done. Most people would try to calculate EV's in blackjack by simulations since it isÂ ...

However, there is a better way to compute the expected value of Ď•(X).... of blackjack that would assure the player a positive expected winning.

## Positive and negative EV (expected value) How to calculate expected value blackjack

## How Expected Value Works in Gambling – Everything You Need to Know How to calculate expected value blackjack

What is expected value, and how does it relate to the game of roulette? Learn how to calculate expected values in roulette games.[request]What is the expected value on a $1 blackjack hand? House rules hit soft 17 apply. Math teacher here. This was asked by a student of mine. I need theÂ ...

In blackjack when an Ace or 10 value card is dealt the casino. If we apply the expected value scenario to a flip of a coin, we know. When we bet 1 dollar per flip, the equation for how much we expect to win over 100 flips is:

## How to calculate expected value blackjack

Introduction Analysis of casino blackjack has been a recurring passion project of mine for nearly two decades now.for blackjack codes matchless have been back at it again for the last couple of months, this time working on computing the distribution and thus also the variance of the outcome of a round.

This post summarizes the initial results of that effort.

For reference up front, all updated source code is availablewith pre-compiled binaries for Windows in the usual location.

Up to this point, the focus has been on and calculation of exact expected value EV of the outcome of a round, for arbitrary shoe compositions and playing strategies, most recently including using a specified card counting system.

But advantage players do not just vary playing strategy, they alsoâ€” and arguably more importantlyâ€” have a betting strategy, wagering more when they perceive how to calculate expected value blackjack shoe to be favorable, wagering less how to calculate expected value blackjack not at all when the shoe is unfavorable.

So a natural next step is to evaluate such betting strategiesâ€¦ but to do so requires not just expected value, but also the of the outcome of each round.

This post describes the software updates for computing not just the variance, but the entire probability distribution of possible outcomes of a round of blackjack.

Rules of the game For consistency in all discussion, examples, click figures, I will assume the following setup: 6 decks dealt to 75% penetration, dealer stands on soft 17, doubling down is allowed on any two cards including after splitting player blackjack, no surrenderâ€¦ and pairs may be split only once i.

Note that this is almost exactly the click setup as the of card counting playing efficiency, with the exception of not re-splitting pairs: although we can efficiently compute exact expected values even when re-splitting pairs is allowed, computing the distribution in that case appears to be much harder.

The return value indicates the action the player should take: whether to stand, hit, double how to calculate expected value blackjack, split, or surrender.

When only computing EV, we can do this efficiently, but for computing the distribution, an explicit specification of playing strategy is required.

The figure below shows a comparison of the resulting distribution of outcomes for these two example strategies.

Algorithm details Blackjack is hard i.

Better yet, when both halves of the split are resolved using the same playing strategy, those individual expected values are equal, so that we need only compute.

When pairs may be split and re-split, things get slightly more complicated, but.

Variance, on the other hand, article source not linearâ€” at least when the summands are correlated, which they certainly are in the case of blackjack split hands.

So my first thought was to simply try brute force, recursively visiting all possible resolutions of a round.

Weighting each resulting outcome by its probability of occurrence, we can compute not just the variance, but the entire distribution of the outcome of the round.

This was unacceptably slow, taking about 5 minutes on my laptop to compute the distribution for CDZ- strategy from a full 6-deck shoe.

Compare this with less than a tenth of a second to compute the CDZ- strategy itself and the corresponding overall EV.

Because computing that distribution involved adding up over half a billion individual possible outcomes of a round, numerical error resulted in loss of nearly half of the digits of double precision.

A simple implementation of cleaned things up.

Two modifications to this initial approach resulted in a roughly 200-fold speedup, so that the current implementation now takes about 2.

This was an interesting trade-off: when you know you need all possible up cards for all possible player handsâ€” as in the case of computing overall EVâ€” there are savings to be gained by doing the computation for all 10 up cards at once.

But when computing the distribution, some player hands are only ever reached with some dealer up cards, so there is benefit in only doing the computation for those that you needâ€¦ even if doing them separately actually makes some of that computation redundant.

The key observation is that 1 both halves of the split have the same set how to calculate expected value blackjack possible handsâ€” viewed as unordered subsets of cardsâ€” that may result; and 2 each possible selection of pairs of such resolved split hands may occur many times, but each with the same probability.

So we need only recursively traverse one half of any particular split hand, recording the number of times we visit each resolved hand subset.

If you are dealt blackjack, then you can do better than an EV of 1.

Although interesting, so far no big deal; the previous and current implementations both handle this situation correctly.

But now suppose that you are initially dealt a pair of tens, and optimal strategy is to split the pair, and you subsequently draw an ace to one of those split hands.

Since this violates the zero memory constraint of CDZ- the current updated implementation performs this correction before post-split EVs are computed, so that the ten-ace playing decision will be the same in both cases.

Note that the strategy calculator also supports the more complex CDP1 and CDP strategies as well, the distinctions of which are how to calculate expected value blackjack another discussion.

What I found interesting was just how common this seemingly pathological situation is.

In a simulation of 100,000 shoes played to 75% penetration, roughly 14.

As expected, these situations invariably occurred very close to the cut card, with depleted shoes still overly rich in tens.

For one specific example, consider the shoe represented by the tuple 6, 6, 1, 5, 8, 2, 5, 9, 2, 34indicating 6 aces, 6 twos, 34 tens, etc.

Wrapping up, this post really just captures my notes on the assumptions and implementation details of the computation of the probability distribution of outcomes of a round of blackjack.

The next step is to look at some actual data, where one issue I want to focus on is a comparison of analysis approachesâ€” combinatorial analysis CA and Monte Carlo simulationâ€” their relative merits, and in particular, the benefits of combining the two approaches where appropriate.

Calculating Expected values and Chi Squared Values

## How Expected Value Works in Gambling – Everything You Need to Know How to calculate expected value blackjack

## Distribution and variance in blackjack – Possibly Wrong How to calculate expected value blackjack

How to calculate the advantage or disadvantage of any random or lucky bet, if you know the probability of winning. It's easy.Expected Value (EV) is the same as expectation and often used interchangeably with the term advantage. May also include the additional value from compsÂ ...

Good Blackjack and Spanish 21 games have house edges below 0.5%. Example #1:. Therefore, the expected value may be calculted as follows: ExpectedÂ ...