JoB Session Variance Dummy

The math is beyond me, and I’m trying my best to understand this. I’ve attempted to write my own program with ChatGPT and also used the session calculator on Wizard of Odds, but I can’t seem to get results that make sense.

Given a video poker game like Jacks or Better (with a known RTP and variance), can I accurately calculate my expected win/loss ranges for different confidence intervals for a given session?

The issue I’m running into:

Both with my ChatGPT-based calculator and the Wizard of Odds session calculator, the results seem to high based on my experience and in the case of my own chatty/python program, indicate that I can lose more than I wager, which doesn’t seem possible.

I’m entering 0.005 for the house edge and 19 for variance—am I misunderstanding how these values should be applied?
Additionally, I’m confused about variance in multi-hand play (e.g., 10-play machines).

I’ve read that variance should be higher for 10-play, but also that multi-hand play reduces variance because the draw variance is smoothed out.
After reading the Wizard of Odds variance page, I’m still not clear on why 10-play variance is higher than single-line, and whether that applies at the session level.
Any help clarifying these two issues—session win/loss calculation and multi-hand variance—would be greatly appreciated!

I don't know the maths of it, but you may never hit on those other 9 hands but hit on one. The best multi-line for least variance is 3 handed, because if you hit 3 of a kind on one, it pays for them all. If you hit a straight on one, you profit. If you hit a flush on one you profit. You may not improve enough to recoup 5 or 10 or more lines with drawing hands.

You MUST hit royals in order to realize the 99.54% return. You are going to go on a steadily downward spiral.

You can play with less of a bankroll for longer on a multi line game, but you are going to lose more and lose more slower than on a single line game.

Me personally, I vote for a little higher denomination on a single line rather than a multi line.

For example.. $2 ($10 bet) single line, rather than 10 line $0.50 ($25 bet).

Take a $8,000 royal rather than a $2,000 royal and $62.50 for 4oak? that is just painful...

Quote: BrestGrambler

...I’ve read that variance should be higher for 10-play, but also that multi-hand play reduces variance because the draw variance is smoothed out.

After reading the Wizard of Odds variance page, I’m still not clear on why 10-play variance is higher than single-line, and whether that applies at the session level.

Any help clarifying these two issues—session win/loss calculation and multi-hand variance—would be greatly appreciated!
link to original post

For the same pay table and same TOTAL AMOUNT BET, a multi-hand game will generate less variance than a single-line game.

For example - A $1 denom 5-play game at max bet ($25 total bet) will have less variance than a $5 denom single-line game at max credits ($25 total bet).

After 20,000 hands on the home VP game of various 1 hand VP games, I'm losing about 10 cents per $1.25 bet, or down $2,000. It takes a Royal to cut that loss in half, two to get back to even.

ChatGPT Deep research on variance differences between single play and multiplayer machines. Apparently a matter of perspective!

Tldr: per coin/hand variance is higher, per total wager variance is lower.

Variance in Single-Play vs. 10-Play Jacks or Better: A Comprehensive Analysis

Introduction

Video poker variance refers to the volatility of outcomes – how much actual results fluctuate around the expected value. In 9/6 Jacks or Better (JoB), a classic low-volatility game, the single-line (one hand at a time) variance is well known to be about 19.5 (per one coin bet). However, opinions diverge when it comes to multi-play machines (such as 10-play, where ten hands are played simultaneously from the same deal). Some sources claim multi-hand games are more volatile because you’re resolving multiple hands at once (implying larger swings), while others argue multi-hand play is less volatile because all hands share the same initial deal, which averages out some of the randomness of the draw.

This report will reconcile these conflicting claims by examining the mathematics of variance in single vs. multi-play 9/6 Jacks or Better. We present expected variance numbers from reputable sources and explain how variance manifests in practice – in terms of bankroll swings and win/loss frequency – for single-line versus 10-play (and other multi-play) JoB. Ultimately, both perspectives are valid; the apparent contradiction arises from different ways of measuring variance. We will clarify these differences and show why multi-hand play can feel “smoother” in practice even as it can produce bigger extreme outcomes.

Understanding Variance in Video Poker

In video poker, variance is a measure of how much the results deviate from the mean. A higher variance game has less frequent but larger wins (bigger swings), while a lower variance game yields more frequent small wins and a steadier bankroll trajectory. For 9/6 Jacks or Better, nearly all payouts are relatively small (with the exception of the rare royal flush), so single-line JoB has a modest variance around 19.5 (per coin bet). This means if you bet one coin repeatedly, the distribution of returns has a variance of ~19.5. In practice, that corresponds to a standard deviation of about 4.4 times your bet, indicating a mild volatility compared to more “top-heavy” paytables (like Bonus or Double Bonus poker which devote more payout to rare four-of-a-kinds, raising variance to 28 or higher).

Crucially, the total variance of a video poker hand can be thought of as two components: the variance from the initial deal and the variance from the subsequent draw. Stewart Ethier (a mathematics professor) and others have shown:

Deal variance: the variability in what the initial 5-card deal gives you (sometimes you’re dealt a pat hand or a high pair, often you’re dealt nothing). For JoB this is relatively small – on the order of ~2 (per coin). You rarely get a huge win immediately on the deal in JoB (the chance of a dealt royal is about 1 in 650,000, which contributes to deal variance, but it’s extremely rare).

Draw variance: the variability in the outcome after you draw (given whatever you held from the deal). This is the larger factor – about ~17.5 in JoB (per coin). The draw adds volatility because, for example, you might hold one high card and sometimes improve to a paying pair or two-pair, but other times you miss entirely. Or you might be one card away from a royal flush: in single-line play you either hit that 1 in 47 chance for a 4,000-coin jackpot or you don’t – a huge swing. In fact, one of the biggest contributors to JoB’s variance is the royal flush cycle on the draw (approximately 1 in 40,000 hands on the draw).


For single-line JoB, these components add up: Variance_total = Variance_deal + Variance_draw ≈ 2 + 17.5 ≈ 19.5. Now, when we move to multi-hand play, these two components are affected differently, which is the key to understanding why variance can be viewed as both higher and lower in multi-play compared to single-play.

Multi-Play Mechanics and Correlated Outcomes

In a multi-hand (multi-play) video poker game – e.g. 10-play JoB – the machine deals one common set of 5 initial cards, and you choose which to hold. Those held cards are then copied across ten separate hands, and the machine draws replacement cards independently for each hand from its own deck. In other words, all hands share the same initial deal, but the draw for each hand is separate. This introduces a correlation between the outcomes of the hands:

If the initial deal is strong, it benefits all hands simultaneously. For example, if you’re dealt a high pair, every one of the 10 hands will at least pay out 1-for-1 (a pair of jacks or better) on the deal, and some hands may improve further on the draw. If you’re dealt a pat flush or full house, all 10 hands return that payout. In the extreme case, if you’re dealt a royal flush, all 10 hands hit the royal – a 10× jackpot win at once. This “all or nothing” aspect of the deal can make multi-play outcomes more swingy when considering the round as a whole, since the initial deal’s strength applies to every line.

If the initial deal is weak (garbage), all hands start from a poor position. Say you have no pay-worthy cards at all and must discard everything – all 10 hands are drawing 5 new cards. In such cases, the chance that none of the 10 hands ends up with a winner is fairly high, meaning you lose all 10 bets in that round. On the other hand, having 10 independent draws also gives you 10 shots to snag a winning combination. It’s more likely at least one of the 10 hands will catch a lucky draw (like pairing up a random hold or hitting a 4-card flush/straight draw) compared to a single hand’s one shot. This tends to smooth out the overall result – often one or a few of the hands will pay something, softening the blow of a bad deal.


In short, multi-play concentrates the effect of the deal (good or bad) across all simultaneous hands, while the draw outcomes introduce some diversification. The net effect is that the outcomes of the hands are positively correlated – not completely independent, but not perfectly tied either. This correlation is why we see different statements about variance:

Perspective 1 – “Higher Variance”: From a bankroll standpoint, if you wager the same denomination per line, a multi-hand game can hit your bankroll harder in a single round (worst-case all hands lose) or boost it dramatically (best-case many hands win big) more than a single-line game could. In other words, the absolute swings (total coins won or lost in one deal) are larger. You’re essentially betting multiple coins at once on one deal, so the variance of the total outcome per round increases with more lines. As one article puts it, “because the strength of each multiple-play game is determined by the initial five cards, if it’s a strong hand, each play will be strong; if it’s a weak hand, each play will be weak.” Thus, more hands = more money riding on that one deal’s luck, which increases total variance. For example, a source calculated that for 9/6 JoB: a 3-play game has variance ~23.4 (about 20% higher than 19.5), 5-play about 27.3 (40% higher), and 100-play about 214 (nearly 11 times higher!). In practical terms, a 100-play machine “we’re talking about a huge variance for a game that initially has relatively low variance” – you could lose 100 bets in one go on a bad deal, or hit an astronomical result on a great deal. If you do not adjust your bet size, multi-line play requires a bigger bankroll to handle the larger swings.

Perspective 2 – “Lower Variance”: From a risk-per-dollar-wagered standpoint, multi-hand play actually makes your results more consistent relative to your total bet. If you treat one round of 10-play (betting 10 coins) as analogous to playing 10 single-hand rounds of 1 coin each, the multi-play scenario will have lower variance in the aggregate outcome than 10 completely independent single hands would. This is because all 10 hands in multi-play share the same deal, effectively averaging out the high-volatility draw component across them. In fact, mathematician Ethier proved that the variance of the average return per hand goes down as the number of hands goes up. For Jacks or Better, the dominant draw-variance (≈17.5) gets divided by the number of hands, while the small deal-variance (~2) remains constant. Thus, the per-hand variance (per coin) for multi-line play is:


\text{Variance}_{\text{per coin, n-play}} = \text{Variance}_{\text{deal}} + \frac{\text{Variance}_{\text{draw}}}{n}

Where n is the number of hands. For 9/6 JoB, plugging in deal ≈2 and draw ≈17.5:

Single-line (n=1): Var ≈ 2 + 17.5/1 = 19.5.

10-play (n=10): Var ≈ 2 + 17.5/10 = 3.75.


Indeed, Bob Dancer reports the variance per $1 bet for 9/6 JoB as dropping to ~3.72 in Ten Play, with Triple Play at ~7.82 and Five Play ~5.48, all much lower than the single-line 19.5. In other words, if you keep the total bet the same, the multi-play version has far less volatility in terms of percentage return on that bet. Dancer emphasizes that dollar-for-dollar, a 10-play game is “much safer” (lower risk of ruin) than an equivalent single-line wager. One way to see this is to imagine betting $50 in one hand versus betting $5 ten times (or ten hands at once with $5 each). One $50 bet on single-line JoB has a much higher chance to swing wildly (hit a jackpot or nothing) than ten $5 bets which will tend to gravitate toward the expected return with less dispersion. Multi-line play, by spreading your wager across multiple hands that share the same hold, brings the actual result closer to the expected value of ~99.5% return, thus reducing volatility per dollar.

Both perspectives are correct; they’re just describing variance in different terms. The key distinction is what is being held constant in the comparison:

Perspective 1 (higher variance) assumes each line’s bet is the same as the single-line bet. For instance, comparing a 25¢ single-line game (5 coins = $1.25 total) to a 25¢ 10-play game (5 coins each line = $12.50 total). Of course the 10-play has 10 times the money at stake per deal – so raw swings in monetary terms are larger (tenfold in this example). This is why intuitively players feel they need a bigger bankroll to play multi-hand at the same denomination. The “variance” figures cited in this context (e.g. 23.4 for 3-play JoB, 27.3 for 5-play, etc.) are usually referring to the effective variance per coin when playing multi-line without scaling down the bet. They rise with n because of the positive correlation between hands (if all 10 coins ride on one deal’s outcome, it’s riskier per coin than if each coin were on independent deals).

Perspective 2 (lower variance) assumes the total bet is the same. For example, compare betting $1.25 on a single hand versus $0.125 on each of 10 hands (still $1.25 total per round). In that scenario, multi-hand play clearly reduces the volatility of your session. The variance per dollar wagered (as a fraction of the total bet) goes down with more hands. In fact, as n→∞, the return per dollar would approach a constant (the deal’s variance contribution plus almost negligible draw variance). The figures cited in this context (e.g. 3.72 for 10-play JoB) represent the variance of the average outcome per coin – essentially the variance of your return rate for one round of play. These numbers drop with n, reflecting the smoothing effect of multiple hands.


To put it succinctly: if you keep coin size fixed, more lines = more total variance; if you keep total wager fixed, more lines = lower variance per dollar. One expert explains it as *“Dollar Ten Play and $10 single play both require $50 to play. Between these two, the Ten Play version has a much lower variance.”*. But if you compared Ten Play $1 vs single-line $1 (so $50 vs $5 total), the Ten Play will feel much swingier in absolute terms.

Expected Variance Numbers for Single vs. Multi-Play JoB

Let’s compile some actual variance figures for full-pay 9/6 JoB from the literature, to cement these concepts:

Single-Line (1 hand): Variance ≈ 19.5 per coin. This is our baseline (standard deviation ≈ 4.4 times the bet).

Triple-Play (3 hands):

Per dollar of total bet: Var ≈ 7.8 (as an average per coin). This corresponds to a standard deviation about 2.8 times the bet – clearly more stable per dollar than single-line.

If coin size is same as single-line: Effective per-coin variance ≈ 23.4 (which is 20% higher than 19.5). In practice, a quarter 3-play machine betting max credits ($1.25 × 3 = $3.75 a deal) will have bigger monetary swings than a single-line quarter machine ($1.25 a deal).


Five-Play (5 hands):

Per dollar (normalized): Var ≈ 5.5. Standard deviation ~2.34 times bet.

Same coin size: Effective per-coin variance ≈ 27.3 (about 40% higher than single-line).


Ten-Play (10 hands):

Per dollar (normalized): Var ≈ 3.7. This is a dramatic reduction – the standard deviation per dollar is only ~√3.7 ≈ 1.93 times the bet. In other words, ten-play JoB returns hover much closer to the expected value on each round, compared to single-play. Bob Dancer notes this as a big bankroll-preservation advantage.

Same coin size: Effective per-coin variance ≈ 37 (roughly; about 90% higher than single-line). While I haven’t seen the 10-play figure explicitly published in the same source, it can be inferred from Wizard of Odds’ covariance calculations. It means if you flat-out bet 10 coins on one deal (via 10-play), the distribution of each coin’s fate is almost twice as volatile as a coin in single-line play due to the all-or-nothing deal effect.


100-Play (for completeness):

Per dollar: Var ≈ 2.15, nearly approaching just the deal variance. With 100 hands, the huge sample of draws averages out so much that your return per round will be very consistent – around 99.5% of your total bet on average, with relatively little fluctuation (standard deviation ~1.47 times the bet).

Same coin: Effective per-coin variance ≈ 214. This eye-popping number reflects that 100 simultaneous bets on one deal can lead to extremely large wins or losses. (E.g. a dealt royal = 100 royals at once, a once-in-a-lifetime $100,000 hand if playing quarters; or a bad run of deals could drain your credits rapidly.)



These numbers illustrate why the disagreement exists: one can quote the 3.7 vs 19.5 (showing 10-play has one-fifth the variance per dollar) or 37 vs 19.5 (showing 10-play has almost double the variance per coin) and both are factually correct. Many gambling resources aimed at casual players focus on the fact that if you sit at a multi-hand machine of the same denomination, you’ll experience larger bankroll swings and need more money to survive, hence “variance is higher.” For example, Casino Player magazine warns that “because over half of all hands in video poker end up losers, the variance for multiple-play games is higher than for single-play games. You need a bigger bankroll to play multi-play games of the same denomination.” By contrast, mathematically inclined analysts (and players who adjust their bet size) will point out that multi-line play reduces variance per unit wagered, making it a safer, less volatile way to wager a given total amount.

How Variance Manifests in Practice

Theory aside, how do these variance differences feel when you’re actually playing? The user’s own observation was that 10-play “feels” less volatile, with more frequent but smaller hits, and this is a common experience. Here’s why:

Frequency of Wins: In single-line JoB, a little over 45% of hands return at least something (pair of jacks or better or higher) – meaning ~55% of hands are total losses. It’s not uncommon to see a long stretch of nothing but losing hands, punctuated by an occasional win (often just your bet back for a high pair). In 10-play, each deal can yield up to 10 winning hands. Even if the initial deal is poor, you have multiple opportunities to pull out a paying result. For example, if you hold a lone high card across 10 hands, the probability that none of the 10 hands pairs it is actually fairly low – usually one or two of those hands will hit a high pair (or better) and pay 1-for-1. This means you’ll rarely see all 10 hands blank out unless the deal was truly hopeless. As one forum participant describes, more lines give “more chances to hit something good, but at the expense of the other losing lines” – in effect you trade the possibility of one big hit for more frequent smaller hits. The result is a steadier drip of payouts. In practice, a 10-play machine will produce some return on a higher percentage of deals compared to a single-line machine, which makes the gameplay feel less brutal during cold streaks.

Magnitude of Wins: When you do hit a great result on single-line, it’s on one hand. A straight flush on a single hand at 5-coin quarters pays 250 coins ($62.50). On a 10-play quarter machine, that same scenario could yield multiple straight flushes if the initial deal set it up. However, typically you might get one or two such premiums among the 10 hands. So instead of one 250-coin win, you might get two 250-coin wins (500 coins total) on that deal – a larger absolute win, but not 10× unless the hand was dealt perfect. More commonly, multi-line play yields many medium wins rather than one huge win. For instance, if you’re dealt a high pair in 10-play, all 10 hands at least start with a winning pair (each pays 1× bet), and some may improve to two-pair or three-of-a-kind. You could end up with, say, 3 hands that improved to two-pair (pay 2× each), 1 hand that made three-of-a-kind (3×), and the rest just a pair (1×). The total might be around 3+3+... = 15 or 16 coins back on a 50-coin wager – a loss overall for that deal, but a much smaller loss than losing 50 coins outright. In single-line, you either would have gotten 1 coin back on a 5-coin bet (if just a pair) or maybe 2 or 3 if it improved – in any case a smaller absolute win but also a larger relative loss (since you bet 5). Multi-hand play tends to cluster payouts, yielding occasional rounds where you hit multiple solid hands at once – making your bankroll jump – but also ensuring that many bad deals still return a fraction of your bet.

Bankroll Swings: Because of the above, a 10-play JoB session typically has a gentler bankroll curve if you’ve adjusted your bet size appropriately. If you play 10-play at one-tenth the denomination of your single-line play (so that your total bet is equal), you will notice fewer extreme downswings. Your bankroll will more often hover near its starting point, with small oscillations, since the variance per dollar is lower. In gambling terms, your risk of ruin is significantly reduced – one analysis notes that playing multi-line “reduces your risk of ruin — i.e., your chance of going broke” compared to single-line, assuming the same payback percentage. This is why advantage players who value longevity often prefer multi-line when a good paytable is available – it lets them cycle more coin with less risk. As a commenter quipped, multi-line results stay “closer to the EV…that’s actually a good thing” for your bankroll health.


On the flip side, if you do not adjust your bet and instead wager max coins on all 10 lines (a much larger total bet), the swings will be correspondingly larger in absolute money. You’ll see big downturns if you hit a patch of bad initial deals (losing $12.50 per round on a quarter 10-play can drain credits quickly), and big spikes if you hit something like a dealt trip that turns into several full houses or quads. In this scenario, the volatility feels higher simply because you’re effectively playing 10 times faster in terms of money at risk. As one source advises, “variance increases as the number of plays increase… You need a bigger bankroll to play multi-play games of the same denomination”. This is why casinos often offer slightly lower paytables on multi-hand games – the lower risk per dollar for the player means a skilled player could sustain long play with less chance of tapping out, so the casino may compensate by reducing the return percentage. (For example, full-pay 9/6 JoB is rarer on 50-play or 100-play machines). Indeed, players have noticed that when the same paytable is available, multi-line is the “safer play,” and speculate that casinos prefer to reduce the payback on those games because of the lower variance advantage to the player.

Reconciling the Conflicting Claims

In summary, the confusion about variance in single vs. 10-play Jacks or Better boils down to context:

Those claiming 10-play has higher variance are usually speaking about the total variance of each deal’s outcome when you play all 10 hands at once with the same stake per hand as a single-line game. In that sense, yes, 10-play will produce bigger bankroll swings (both positive and negative) than one hand of single-line, since you’ve effectively multiplied the variance by adding correlated bets. All 10 hands win or lose together to a large extent, making the distribution of outcomes wider. For example, being dealt a rare hand (like four of a kind dealt, or a straight flush dealt) yields 10 simultaneous winners – an outcome unattainable in one-hand play – which pushes the variance upward. These sources are emphasizing bankroll requirements and volatility of session results when playing multi-line without scaling down. In practical advice: if you move from one-line quarters to 10-line quarters, expect a bumpy ride and bring more bankroll to survive the dry spells.

The others saying 10-play has lower variance are focusing on the variance per coin or per dollar wagered. They consider that if you’re going to bet, say, $5 per deal, you can do it as one hand at $5 or 10 hands at $0.50 each – and the latter will yield more stable returns. Statistically, multi-play greatly reduces the variance of the draw (the main source of volatility) by averaging it over many hands. These sources talk about risk of ruin, return to player consistency, and “feeling” of volatility. From this angle, 10-play is indeed a smoother, lower-variance way to play $X total bet – you’ll get a lot more small paybacks and avoid long streaks of zero, keeping you closer to the theoretical payback in the short run.


When properly understood, there is no real contradiction: it’s all about whether you treat 10 simultaneous hands as 10 separate bets or as one big bet broken into 10 parts. Multi-play lets you spread out one large bet into many smaller, correlated bets. This diversification reduces relative variance while increasing absolute variance. A succinct forum reply captured it well: *“You’ll see lower variance in the game with more lines if you bet the same amount per round… More lines lead to more chances to hit something good (smoothing out results).”* Conversely, if you don’t adjust your bet per line, “more lines…[lead] to a few trips to variance-ville,” another player joked, meaning bigger swings.

Conclusion

For full-pay 9/6 Jacks or Better, a single-play machine has a variance of ~19.5 (per coin), whereas a 10-play machine can be viewed in two ways: as a high-variance 10-in-1 game (with an effective per-coin variance of ~37 when each hand is played at the same stake), or as a low-variance way to spread a single bet across 10 hands (variance ~3.7 per coin when normalized to the total bet). In practice, many players experience 10-play as less volatile because they typically lower their bet per hand (e.g. playing nickels or dimes on multi-line instead of quarters or dollars on single-line) and enjoy more frequent minor wins. The bankroll lasts longer with multi-hand play due to the smoothing effect of the common deal and multiple draws, even though the potential for extreme outcomes (like hitting a monster hand on all lines) is higher.

To resolve the conflicting claims: both are true. Multi-play increases the swinginess of each deal’s total outcome but decreases the swinginess of your results relative to money wagered. If your goal is to minimize volatility and preserve bankroll, you should choose multi-hand play and adjust your coin size downward so that your total bet remains within your budget – this will give you a steadier ride with Jacks or Better. If instead you’re chasing a big thrill and don’t mind larger bankroll fluctuations, playing a single line at a higher denomination (or playing multi-line without reducing denomination) will provide more dramatic ups and downs.

In summary, for 9/6 JoB: a 10-play game is “safer” for the same total wager (lower risk of ruin, more frequent small payouts), but it can be more volatile in absolute terms if you treat it as ten times the bet (because multiple hands can all win or lose together). Understanding this nuance explains why experts sometimes seem to disagree on multi-hand variance. Always consider whether advice is talking about variance per coin or variance per round, and adjust your bankroll strategy accordingly.

Sources:

Michael Shackelford, Wizard of Odds – Standard Deviation of n-Play Video Poker (variance and covariance for multi-hand JoB)

Bob Dancer, Las Vegas Advisor – Why is the Variance of Multi-Line Play Less than the Variance of Single-Line Play? (Oct 2024)

Casino Player ) – The Tough Truth About Multiple-Play Video Poker

Vegas Fanatics forum – discussion of multi-hand vs single-hand variance (2016)

forum – variance data from Video Poker for Winners software (multi-hand variance numbers for various games)

Wizard of Odds – JoB strategy/return tables and variance info.