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The Sunday Reader - Issue 4: Understanding Odds

Odds are a hot topic in Gamesville blog posts. They can be a bit confusing, so I decided to try and explain some of this stuff. Keep in mind that I majored in English, not math. If you see any boo-boos please let me know! ^_^

The first thing to remember is that odds are very useful for large groups, but are practically meaningless for individuals. Seriously!

Perhaps you've heard (or said) this before. "I've played for YEARS and not won anything! Yet this new person played for only a few months before she won something. It has to be rigged!"

This type of thinking implies that odds function like a countdown, and it's a very common misconception! It seems logical that someone playing for a long time would win before someone who just started. But odds just don't work that way.

Let's look at some real-world odds for perspective. Did you know that the odds of having an ER worthy accident with a drinking fountain in any year is 1 in 220,700?

If odds worked as a countdown, you'd have to use a drinking fountain 605 times a day for a year in order to "trigger" an accident on your 220,700th visit. That's about 2 visits every 5 minutes for an entire year!

I don't know anyone that thirsty. ^_^

Clearly, odds do not work as a countdown. And unfortunately, people end up in the hospital every year for drinking fountain mishaps. So what gives?

Odds are calculated by taking the number of people (or attempts) and dividing by the number of incidents. In this case, there are approximately 307 million people in the United States. And every year, roughly 1391 people in the US wind up at the ER from a drinking fountain accident.

307 million people divided by 1391 fountain accidents = 220,700. Which is why the odds are 1 in 220,700 that a person in the United States will have an accident.

There are ways to affect your individual odds. If you avoid drinking fountains for the rest of your life, your personal odds are pretty close to zero. (I'm not going to discount the possibility of a drinking fountain falling from the sky and bonking you on the head. Weirder things have happened.)

Meanwhile, if you're feeling masochistic you could use drinking fountains as often as possible. Then you'd have a better chance of having an accident versus the person living in a water-fountain-free bomb shelter. That part makes sense. ^_^

HOWEVER, every time you use a drinking fountain, you still have a 1 in 220,700 chance of having an accident. It doesn't matter if you visit one every day or once a year. Someone could have an accident on their first trip!

Now let's apply this to games.

Cindy, Jan, and Marsha head to an arcade and they all play the same game. The game isn't rigged to favor any particular person, but here's what happens:
  • Cindy plays 500 times and wins on her 500th play. Woohoo!
  • Jan plays 10 times and wins on her 2nd and 10th play!! (Lucky girl!)
  • Marsha plays 2,490 times and doesn't win anything. :(
Can you figure out the odds for this game?

First we count the total number of plays. (3,000 in this case.) Then we count the number of wins. (1 for Cindy, 2 for Jan, 0 for Marsha. 3 total wins.) Now we divide the number of plays by the number of wins. 3/3000 = 1/1000. Each time someone sits down to play they have a 1 in 1,000 shot of winning.

But wait! Marsha played 2,490 times! If the odds are 1 in 1,000 then she should have won at least twice, right?

Nope! This is why I say odds are practically meaningless when applied to individual people. Remember, odds are calculated by using group results! That makes a huge difference!

If Marsha was the only person to ever play that game, then yeah, she would eventually win. She might also play 5,000+ times before she won anything, and then have several wins in quick succession. Odds don't care about who you are or how long you've been playing.

Now here comes the next confusing part: Understanding how some people can win multiple times.

Bobby, Peter, and Greg hear about Jan's crazy luck so they go play the exact same game with the 1 in 1,000 odds. Each of them play exactly 1,000 times. (It was a slow day.) They thought that each of them would win once, but that didn't happen. Instead, Greg won twice and Peter didn't win anything at all!

As you might imagine, life at the Brady house turns chaotic. Marsha is certain that Jan cheated, and Peter stomps around the house loudly saying, "I NEVER win!!"

Finally it reaches a breaking point. Carol, Mike, and Alice all head down to the arcade and demand to speak to the manager. They're sure he's rigging the games somehow and they'll call the police, gosh darn it!

With a sigh, the manager produces a coin. He says, "Each time you flip a coin you have a 50% chance of the coin landing heads up. (1 in 2 odds.) If you flip a coin 10 times you probably expect to end up with 5 heads and 5 tails." They nod and agree that sounds right.

He digs in his pockets to get more coins, and then asks each of them to flip a coin 10 times to see what happens. (Alice is suspicious and uses one of her own coins.) Here were their results.
  • Carol flipped 6 heads and 4 tails
  • Mike flipped 6 heads and 4 tails
  • Alice flipped 2 heads and 8 tails
All three were stunned. It should have been even!

The manager pulls out a white board and starts to explain combinatorics. Combinatorics looks at combinations (duh) and probability. ^_^

In the case of 10 penny flips, there are 11 possible combinations. (Edit: I oversimplified this. Read Chris's comment at the end for a full explanation!)
  • 0 heads, 10 tails
  • 1 head, 9 tails
  • 2 heads, 8 tails
  • 3 heads, 7 tails
  • 4 heads, 6 tails
  • 5 heads, 5 tails
  • 6 heads, 4 tails
  • 7 heads, 3 tails
  • 8 heads, 2 tails
  • 9 heads, 1 tail
  • 10 heads, 0 tails

This means that the odds of flipping exactly 5 heads and 5 tails are 1 in 11. (Or a 9% chance.) This also means that you have a 10 in 11 chance of flipping any other combination. That's a 91% chance of getting something other than 5 heads 5 tails!

You can try this at home too! I think it's fascinating. ^_^

But how does combinatorics relate to winning games? Well, one of the biggest complaints at Gamesville is that there are repeat winners. Frankly, it would be suspect if there weren't any repeat winners! Let's take a look at Bobby, Peter, and Greg again. They were playing a 1 in 1,000 odds game, so it was pretty likely that they would have 3 wins in their 3,000 total plays. How likely is it that each of them would win one prize?

Let's see! In the graph below, each line across represents one possible outcome.

Bobby Peter Greg
1 1 1
2 1 0
2 0 1
1 2 0
0 2 1
1 0 2
0 1 2
3 0 0
0 3 0
0 0 3

Assuming that 3 prizes are given out, there are ten total combinations. That makes percents easy. Woohoo!
  • 10% chance that each player will win one prize each.
  • 30% chance that one player will win ALL of the prizes!
  • 60% chance that one player will win two prizes and one player gets nothing.
It's entirely possible that each player could get one prize each. However, it is the least likely combination! And if you didn't notice, there is a 90% chance that at least one player will not win anything.

This is a simplified explanation of combinatorics, but hopefully you get the idea. ^_^

I'd like to end this post with something that I read in a news article many years ago. It was a study on people who defined themselves as lucky versus people who defined themselves as unlucky. Unsurprisingly it turned out that the lucky people really did have better outcomes! The researchers looked closer to try and figure out what made one person lucky over another. And you know what the biggest difference is? Lucky people keep trying. Unlucky people generally give up quicker.

There are more prizes on Gamesville than any other free site that I've found, and you do have better odds of winning something at Gamesville versus, say, iWon. Regardless, you are never guaranteed a win. It's entirely possible to play for years and never win anything. But if you stop playing entirely, then your odds are zero. You've no chance of winning if you don't play!

I hope you enjoyed this installment of the Sunday Reader. ^_^ If you have any questions or comments, please post!

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