GamblingWhat are the odds?

Jessica Judd

Slot machinesSlot machines are designed

with three or more reels decorated with a specific number of symbols, and

the object of the game is to get matching symbols after

all reels have spun.

If the arrangement is 3 reels with 10 symbols on

each…

What are the odds?

ODDSAccording to the Theoretical Method of Probability:

The odds of any particular symbol hitting on a single reel would be 1 out of 10, or 1/10.

The odds of 2 of the same symbol hitting on a spin would be 1/10 × 1/10, or 1/100.

The odds of 3 of the same symbol hitting on a spin would be 1/10 × 1/10 × 1/10, or 1/1000.

making the odds 999 to 1

However,The modern slot machine nolonger runs on physical reels,but rather mechanical reels

have been replaced withvideo screens.

The odds of video slotmachines are no longer

based on a certain numberhitting but are rather run byrandom number generators

and the modernizedmachines have drastically

changed how the odds work.

oddsWhile a physical reel can only hold about 20 symbols before becoming too big to be practical, a video slot

machine could, theoretically, have 100 symbols per reel, and although a physical machine can't really accommodate

more than 3 reels comfortably, a video slot machine has no such limitations, and could have 5 reels or more.

Assuming that the theoretical method could still apply to the new machines, having 100 symbols per 5 reels takes the

odds from

999 to 1 to 999999999 to 1 (1/100 × 1/100 × 1/100 × 1/100 × 1/100 =

1/10000000000.)

Random Number Generators

To make things even more interesting, the random number generator (RNG) can be, and is, programmed to

give certain symbols a different chance of coming up that is totally unrelated to the number of symbols,

Meaning the theoretical method that was used before is completely inaccurate.

The slot machine might have one cherry out of ten symbols, but the generator might be programmed to only have that cherry land once out of every twenty

random spins,

potentially making the odds EVEN SMALLER than 999999999 to 1.

In conclusionThe result of the RNG is that

combinations can be programmed to be less likely, and the less likely it is to hit

those combinations, the more the casino can afford to pay out for them on the slight chance that they occur.

Progressive jackpots of six and seven figures would have been impossible on any

mechanical slot machine, but they're quite easy to program

into a video slot machine's random number generator.

The change in how the winners are determined makes these machines

nearly impossible to win, but incredibly profitable (for the

casino) in spite of the new, bigger jackpots.

CrapsThe game is played by the rolling of two dice, and is entirely dependent on the sum of the two dice

after they are rolled. If a player rolls a total of 7 or 11 on the first roll,

they win. If a player rolls a total of 2, 3 or 12 on the first

roll, they lose. If any other number (4, 5, 6, 8, 9, or 10) is

reached on the first roll, that number becomes the “point” and the player must strive to reach that point a second time BEFORE rolling a 7 to

win.

chances of winningAs you can see, there are 36

possible outcomes of rolling two dice.

Going back to the theoretical method of probability, the odds of rolling a 7 are the most likely of all at 6/36

(1/6)The odds of rolling an 11 are

2/36 (1/18.) So the odds of rolling EITHER a

7 or an 11 are 8/36 (2/9) which also equates to 22%. So there is a 22% chance of

winning on the first roll.

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10

11

6 7 8 9 10

11

12

each possible outcome of rolling two dice

chances of losingwhat are the chances of

losing on the first roll by rolling a 2,3 or 12? 2 and 12 both occur once on the chart and

3occurs twice making 4 opportunities to lose

out of 36 (1/9) resulting in an 11% likelihood of losing on

the first roll.

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10

11

6 7 8 9 10

11

12

each possible outcome of rolling two dice

Other possibilitiesBetween the numbers 4, 5, 6, 8, 9

and 10 there is a 24/36 (2/3, 67%) chance of rolling a neutral

number and gaining a point,but the probability of reaching

that number again before reaching a 7 would depend on

what number you rolled. Both 4 and 10 have a 1/12 (3/36,

8.3%) chance of being rolled in comparison the 1/6 of a 7,

5 and 9 share the likelihood of 1/9 (4/36, 11%),

and 6 and 8 are the most likely to achieve for a second time with

5/36 (13.8%.)

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10

11

6 7 8 9 10

11

12

each possible outcome of rolling two dice

In conclusionOverall, 7 is obviously the most likely

number to roll, which plays to the shooter’s advantage on the first roll,

but could be his demise any time after that, making the game of craps as close to a 50/50 shot of winning or

losing as you can get at a casino.

Black Jack (21) Blackjack is the most widely played casino banking game in the world, as

well as the most studied. It is believed to be the one game with

TRUE probability, therefore it is more a game of strategy and math than a

game of chance, unlike most gambling games.

playThe game is played by dealing cards to both the player(s)

and the dealer. Play starts with a card dealt to the player, and then to the

dealer, and then repeating; resulting in both sides possessing a two card hand.

The cards numbered 2-10 are valued at their written number, and all face cards are worth 10 (excluding the ace

which has the value of either 1 or 11 according to the players best interest,)

The object of the game is to get as close to a total of 21 as possible without going over (busting) as well as scoring

higher than the dealer. Each turn consists of the option to be dealt another card (hit) in order to get closer to the

goal of 21.

House advantageIn this game, the house (casino) has the advantage purely just by being the last player to act on every turn. By acting last, this means that every other player has already made their decision and could potentially bust before the house has even played.

Odds of winningTo start, what is the likelihood of actually being dealt a blackjack hand, or an exact sum of 21 consisting of one

ace and one card with the value of 10? Using the combinations formula, we know that there are

1326 different two-card hands in one deck of cards

. In addition to that, we also know that there are 4 possible

aces and 16 possible face cards in that deck, so there are (4 × 16) 64 ways to be dealt a blackjack hand,

out of 1326 64/1326 or a 4.8% chance of winning straight off of the

hand you’re dealt.

oddsUsing this same

principle we are able to determine the odds of any

particular hand being dealt by calculating all the possible ways to achieve

each total.Following this chart you will

see that the most common two card hand, at 38.7%, is a hand totaling 1-16, which is considered

a decision hand.

Two Card Count % Frequency

Natural 21 4.8%

Hard Standing (17-20)

30.0%

Decision Hands (1-16)

38.7%

oddsIn knowing these

odds, and using the same method

of calculating what cards are left in the deck, we can also establish the odds

of busting with each possible

hand.

Hand Value % Bust if you hit

21 100%

20 92%

19 85%

18 77%

17 69%

16 62%

15 58%

14 56%

13 39%

12 41%

11 and under 0%

The rest of the game depends entirely on the “luck of the draw”, but regardless of the outcome, you can use probability to figure out what your next

move should be.

An example:Suppose you are dealt a 4 and a 9,

resulting in a total sum of 13.

We know there are 50 cards left in the deck:

52 - the 2 in your hand = 50

An example:16 of those 50 cards have a value of ten (all

face cards and tens) -which would result in a bust.

13 + 10 = 23 as would all values greater than 8

13 + 8 = 21 which accounts for 19 out of the 50 cards

16 face cards/tens + 3 nines = 19leaving you with a 38% (19/50) chance for a

bust.

An example:On the other hand, all values less than or equal

to 8(including Aces, which would count as 1 during this hand) account for 31 of the 50

cards, 4 eights + 4 sevens + 4 sixes + 4 fives + 3

fours + 4 threes + 4 twos + 4 aces = 31resulting in a 62% (31/50)chance of being able

to continue.

In this case, it wouldn’t be a bad idea to draw another card.

Counting CardsThis is also the idea behind counting cards; to be able to track what cards

are left in the deck to determine what is most likely going to be the

next card dealt, but it gets complicated as it is hard to keep track of all cards dealt to other

players as well.

In ConclusionContrary to the popular belief that if you’ve been losing all

day you’re “due” for a win, the reality is that all plays are independent, and have nothing to do with each other.

The calculated statistics are pretty accurate in the long run, but only the guy that’s been winning all day is the one

accounting for the win you’re “due” for

So, despite the casino’s way of luring you in, it is possible to win, and some games are more fair than others, but having the knowledge of the math behind it all could

save everyone at the tables and machines a lot of money.