Hit frequency or chance of winning something

I was recently told by a game inventor that the chances of winning at least something (maybe not a net profit, but something back from original wagers) on a game was 70+%. I can't figure out how to figure out the math on it. Some people will win bet 1 but not 2 or 3. Some will win bet 3 but not 1 or 2. I don't think you add the three percentages and divide by 3.

So if the following is true:

Bet 1 - 48.93% hit frequency
Bet 2 - 24.79% hit frequency
Bet 3 - 19.72% hit frequency

Is there an answer to this with just this limited information??



ZCore13

There's no way of knowing without knowing exactly how the three bets work.

How is it possible for some people to win bet 3 but not 1 or 2, and some people to win bet 1 but not 2 or 3?
Are these two groups of people involved in two entirely different bets?
Is it possible to win more than one bet at a time?

If the three bets are entirely different things - i.e. there is no chance to win two or three bets at the same time - then the probability of winning "something" is the sum of the three, or 93.44%.

The easiest way to explain it: suppose you're playing a 4-digit daily lottery, where you choose one of the 10,000 numbers from 0000 to 9999.
"Bet 1" wins if the winning number is anywhere from 0001 to 4893 - a 48.93% chance.
"Bet 2" wins if the winning number is anywhere from 4894 to 7372 - a 24.79% chance.
"Bet 3" wins if the winning number is anywhere from 7393 to 9344 - a 19.72% chance.
You win one of the three bets if the number is anywhere from 0001 to 9344 - a 93.44% chance.


Your question about winning "at least something" reminds me of a mass mailing scam that I haven't seen in a while, where you are guaranteed one of four prizes - usually something like a new car, $5000 cash, a trip to somewhere in Europe, and a $250 gift certificate which turns out to be good only on cheap knockoff luggage which you can probably find for $10 if you look hard enough, but the scam lists as "normally $400, but it's only $150 after you apply your gift certificate".
One company figured that everybody got wise to this, so they put a spin on it, and the postcards started saying that you won TWO prizes from the list.

Yeah, I didn't think just the hit frequencies were enough. Players can win any combination of the three bets. None, one of any of them, two out of three or all. The three bets are independent of each other, but all three are mandatory.


ZCore13

Some UK fruit machines have a reasonable chance of something back, but they do this by having (say) a bet of 50p and offering prizes of 5p, 10p etc. In the US they can do a similar idea by having multiple lines (say on a three reel machine, playing middle line, top/bottom line, each cross line).

If your case I'm guessing it could be Bet 1 is same colour, Bet 2 is same suit, Bet 3 (not sure as 1/5th!).
or (I haven't done the maths but think Three-Card poker, winner takes all)
Bet 1 - suppose you were playing a simple game where the dealer has a slight advantage and wins if you beat the dealer (assuming there is a tie break mechanism or low chance of a tie).
Bet 2 is your first card matches the dealer's upcard (or something similar such as three cards the same colour where p~=1/4).
Bet 3 is (no idea) but might be pair plus like concept.

This is why people like 3-card - although the pair-plus has a relation to the basic game, the prime (colours match) doesn't (except flush=colours match).

Quote: Zcore13

Yeah, I didn't think just the hit frequencies were enough. Players can win any combination of the three bets. None, one of any of them, two out of three or all. The three bets are independent of each other, but all three are mandatory.

Actually, if they are independent, then you can.

Bet 1 has a 51.07% chance of losing
Bet 2 has a 75.21% chance of losing
Bet 3 has a 80.28% chance of losing
If they are independent, then the chance of all three losing is 0.5107 x 0.7521 x 0.8028 = about 0.3084, or 30.84% and the chance of winning at least one of them is 100% - 30.84% = 69.16%

Quote: ThatDonGuy

Actually, if they are independent, then you can.

Bet 1 has a 51.07% chance of losing
Bet 2 has a 75.21% chance of losing
Bet 3 has a 80.28% chance of losing
If they are independent, then the chance of all three losing is 0.5107 x 0.7521 x 0.8028 = about 0.3084, or 30.84% and the chance of winning at least one of them is 100% - 30.84% = 69.16%

Well then maybe the guy was right. He said 70% chance of winning at least one.


ZCore13

I just did a game where you could win any 1 of three bets, any 2, or all 3, and the overall hit rate for at least a partial was 64%. So it can be done.

Z, do you think from a player's perspective, winning something, but still net losing on a hand is a good design feature? I would think that it would be more difficult to record a "net win" for any one seesion in a game designed that way. Wouldn't players would walk away a loser at the game more often than a 42% hit rate type game.....but I would have to have a brighter math mind than me determine that distribution.

Maybe that is OK as regular players expect to lose, but I would be turned off from a game I never felt I could win at, net. In fact I have started to avoid UTH and play more BJ just based on my recent negative volatility at UTH.....I just can't seem to get up at that game anymore, but BJ has been delivering some winning sessions.

Quote: Paradigm

Z, do you think from a player's perspective, winning something, but still net losing on a hand is a good design feature?

Roulette has worked this way for hundreds of years...

Quote: MathExtremist

Roulette has worked this way for hundreds of years...

Yup. People spread $40 around the board
and win $25 and act like they've
accomplished something.

I'm okay with a high hit rate on slot machines.

Frankly, I don't ever expect to actually win anything on them; I expect to be moderately entertained by blinking lights and loud noises as my cash disappears in a slow smolder. (I admit it, I'm guilty of letting a slot machine ring for over half an hour once on a pretty huge win [in terms of credits].)

Quote: EvenBob

Yup. People spread $40 around the board
and win $25 and act like they've
accomplished something.

You & ME are right, I am thinking that spreading on only three bets and winning even (or slighty more) money on one is going to feel different than Roulette with its various bets & odd payouts. What if Roulette was a reduced to only a three Color bet.....you can bet Red, Black & Green only......I think that version of craps would have a tougher time succeeding. I guess I need more info on the three bets to have a better understanding of game flow.

One of the most important game changes in the past 30 years was the widespread behavior of slot games paying back less than 100% of the wager. Traditional 3-reel mechanical games had integer paytables and a single payline. Same with video poker. In both cases, either you lose all of your bet or you get back some integer multiple thereof. With MLMC games you can bet 100 coins and get 20 back. It's a loss of 80 coins but it doesn't feel like a loss. So it's important psychologically, but it's also important mathematically because the outcome distribution now has a big hump between 0 and 1 as opposed to a big gap. That's reflected in what casino operators (and game vendors) call "time on device". The variance is much lower compared to the total bet than in prior games.

These partial losses have also been called "losses disguised as wins" or LDWs, by at least a one academic in research papers. I've read a number of papers written by psychologists on gambling games. Many of them base their experiments on unrealistic simulations of actual casino games but I'm not sure how much that actually matters in the end.

There are artificial ways to accomplish this in certain table games too. In craps or roulette, make slightly unequal bet amounts on opposing wagers, like $100 Red + $105 Black, or $100 Don't + $105 Pass. The variance of such combinations compared to the total wager size is much, much lower than making a single wager for the total amount. In fact, adjusting the variance of a play session is primarily what betting systems accomplish (since we all know they can't actually impact the edge).

By the way, as a result of partial losses, there is now a meaningful distinction between "hit frequency" and "chance of winning something." When I was at Silicon Gaming, we used to approximate two statistics. One was hit frequency and one was win frequency, where hit frequency was a return of >0 and win frequency was a return of >=1x wager (technically, win-or-push frequency). Those used to be the same for most games but now there's a huge difference between how often you get something back and how often you get back more than you wagered. (Operators never seemed to care about the distinction and I don't think it ever really took off in the industry. Modern sell sheets rarely go into much statistical detail about frequencies; rather, they use qualitative phrases like "high volatility".)

Quote: Paradigm

Z, do you think from a player's perspective, winning something, but still net losing on a hand is a good design feature? I would think that it would be more difficult to record a "net win" for any one seesion in a game designed that way. Wouldn't players would walk away a loser at the game more often than a 42% hit rate type game.....but I would have to have a brighter math mind than me determine that distribution.

Maybe that is OK as regular players expect to lose, but I would be turned off from a game I never felt I could win at, net. In fact I have started to avoid UTH and play more BJ just based on my recent negative volatility at UTH.....I just can't seem to get up at that game anymore, but BJ has been delivering some winning sessions.

Slot machines are famous for this tactic. Bet 150 credits, win back 100. Bet 150 credits win nothing. Bet 150 credits win back 250. Bet 150 credits win back 75. Hit percentage 75%. Net loss 175 credits. I believe they are referred to as "trickle back" slot machines. Frog Prince was a hugely popular one.


ZCore13

I'll be posting information on the game I'm referring to today or tomorrow. I'm the 2nd ever installation and I replaced Three Card Poker with it.


ZCore13

Quote: Zcore13

I'll be posting information on the game I'm referring to today or tomorrow. I'm the 2nd ever installation and I replaced Three Card Poker with it.


ZCore13

Looking forward to it.

Quote: Zcore13

Slot machines are famous for this tactic. Bet 150 credits, win back 100. Bet 150 credits win nothing. Bet 150 credits win back 250. Bet 150 credits win back 75. Hit percentage 75%. Net loss 175 credits. I believe they are referred to as "trickle back" slot machines. Frog Prince was a hugely popular one.


ZCore13

The only reason that the slot machines can even get away with calling it a, "Win," I would venture to guess, is that the Pays are on a For-One basis, so the money is presumed lost as soon as it is bet. Think about how much different the slot machine industry would be if the slots were required to say what really happened, like in your example:

LOSE: 50 Credits, LOSE: 150 Credits, WIN: 100 Credits, LOSE: 75 Credits

Players would see how frequently that they are actually losing, and I think most of these multi-line machines would fall out of favor in pretty short order.