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Double or Nothing: Is the Coin Flip Legit?


by lithoxide

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     “Fancy a game of chance?” he drawls, beckoning you closer. You glance over, intrigued. You came to Meridell Castle to flub some jokes for King Skarl. But yet the Skeith calling to you from the treasure room has captured your interest. What could he want, sitting in that treasury surrounded by the mountains of gold? You are drawn in, and before you know it, you’re engaged in a game of chance.

     The rules of Double or Nothing are simple. Wager 10 NP and you participate in a simple game. Snargan the treasurer will flip a coin. If the coin lands on tails, you lose the NP you have wagered. But if chance grants heads, you will double your current wager. You then can choose: continue risking the coin’s fate to double up again, or walk away with what you have earned.

     This game lacks much risk. After all, the most you would lose each tails is the original 10 NP you wagered. That being said, you probably won’t be playing this game for the “immense riches.” You would rather want to play for the chance of winning this shiny new avatar:

     

     
Snargan ain’t much to look at, but he’s trying his best.

     Double or Nothing rewards the avatar once you have collected your winnings when the pot reaches 320 NP. That means winning 5 rounds of the game, the probability of which is relatively low. We know this through the beauty of statistics.

     

     
Your abandoned math homework is calling your name, by the way.

     We all know the probability of a coin flip: 50% chance of heads and 50% chance of tails. When Snargan flips that coin, you have equal probability of advancing to the next round. But here’s the thing with statistics: when we want multiple probabilities to happen in a row, the overall success rate drops considerably. Want to know a trick for calculating this?

     First, we take the probability of the outcome, expressed it in a fraction. In this case, a coin flip will have 2 outcomes: heads and tails. We’re hoping for 1 particular outcome: heads. That is ½ outcome, or a 50% chance.

     Next, figure out how many times we want to see this outcome occur. As we need to win 5 rounds to achieve the avatar, we will need 5 outcomes of heads. That’s 5 different outcomes of ½.

     Lastly, we are going to multiply each of those outcomes together. Multiplying fractions is a straightforward process, especially in a case like this. Five outcomes means that we will have:

     ½ x ½ x ½ x ½ x ½

     We first multiply across the numerators, the numbers above the line. As the numerators are all 1’s, the final numerator will be 1. Then we will multiple across the denominators, the numbers below the line. 2 x 2 x 2 x 2 x 2 = 32, which becomes our final denominator. And finally, we put it all together to create our final probability: 1/32. The probability of the coin landing on heads 5 times in a row is 1/32, or 3.125%. That means that 1 out of every 32 games of Double or Nothing – 3.125% of games – will result in the avatar.

     

     
Math class is over. You can go back to sleep.

     Now it’s true that you could get lucky and win the avatar on your first play. What’s great about statistics is that luck doesn’t play a role in probability. By theory, the avatar is entirely obtainable through sheer perseverance. However, it all depends on whether Snargan plays a fair game. After all, he is a shifty character who is risking gambling with King Skarl’s money. It isn’t in Snargan’s best interests to lose. Can we be sure that he is using a standard issue Skarl-faced coin?

     Being the statistics nerd that I am, I wanted to test Snargan’s fairness in the game. That is why I sat down with him and played 150 rounds of Double or Nothing. It was a simple data collection approach. I kept log of the round’s outcome (win or lose) for each round that I played, as well as how many times I made it to that round. That yielded some telling statistics about Snargan’s operation.

     How did it all turn out? Let’s talk about it round-by-round. Keep in mind that while I should win each individual round 50% of the time, my odds of winning each successive round will drop. (I have a 50% chance of winning Round 1, a 25% chance of winning Round 2, etc.) I’ve reported the suggested probability of winning each round, and checked whether the outcomes match what statistics would suggest!

     Round 1

     
Suggested Probability of Winning: 50%

     
Winnings: 20 NP

     This is the most straightforward round, as you will always start with Round 1. Given this, I played Round 1 150 times, and won in 77 of the games. That’s a win rate of 51.3%, giving Snargan some credibility. But one round isn’t merely enough!

     Round 2

     
Suggested Probability of Winning: 25%

     
Winnings: 40 NP

     I entered Round 2 77 times and I won 40 times. That’s a win rate of 51.9%.

     However, remember that winning Round 2 means that I have gotten heads twice in a row. That’s a probability of ½ x ½, or ¼, 25%. How can we determine whether Snargan is respecting statistics? It’s actually easy, and we can use this method to evaluate each round moving forward! To determine the overall win rate for the Round 2, I take the number of wins in Round 2 (40) and divide it by the total number of games played (150). 40/150 = 27%, which is a bit over our suggested probability of winning. So far, Snargan seems to be playing fair.

     Round 3

     
Suggested Probability of Winning: 12.5%

     
Winnings: 80 NP

     I entered this round 40 times and won 22, which comes out to a 55% win-rate. Calculating the overall win-rate (22/150) shows a 15% win-rate for Round 3.

     Round 4

     
Suggested Probability of Winning: 6.25%

     
Winnings: 160 NP

     I entered this round 22 times and won 13, which comes out to a 59% win-rate. Calculating the overall win-rate (13/150) shows a 9% win-rate for Round 4.

     Round 5: Avatar Round

     
Suggested Probability of Winning: 3.125%

     
Winnings: 320 NP

     This is it: winning this round means securing the avatar! Remember that achieving this avatar means winning the round and collecting your winnings, which in theory should be a 50% shot. However, Snargan seems to finally be showing his shady nature here, because the statistics don’t add up.

     I entered this round 13 times, yet I only won 3 times. This is a win rate of 15.4%, which is far below the 50% probability that should be expected. I do need to note a disclaimer here. Given how statistics works, 13 times is too few to draw a reasonable conclusion. Had I gotten more times in Round 5, the win-rate could have risen to the standard 50%. However, I’m inclined to cast a side eye at Snargan. My previous rounds showed that my win-rate was above what was expected. Yet my win-rate dropped drastically upon entering Round 5. As this is the avatar round, Snargan may be tipping the odds in his favor to keep you from winning.

     

     
Or maybe he’s lonely and he just enjoys the company.

     Calculating the overall win-rate ends up with a 1.33% chance of winning. That’s definitely below the 3.125% rate that statistics would suggest, but the difference is small. Snargan could potentially be shady, but the overall chance of winning the avatar is straightforward.

     From what my statistics suggest, it’ll take 100 games of Double or Nothing to win the avatar. Of course, you may win the avatar much earlier. (I cleared Round 5 on my 15th game in collecting this data, so it can happen much sooner.) If it takes time, however, just stick with it! Probability means that the outcome will eventually happen just so long as you continue playing!

     ”But what if I want more than just the avatar?”

     The avatar isn’t the only benefit you can win from Double or Nothing. You could also try to go for the Double or Nothing trophy! Of course, the best time to try for any trophy is at the beginning of the month during the high-scores table reset. But if you are bold and heading for the trophy during the month, be prepared to show some patience and perseverance. A bronze trophy needs a score of 1,280 or 2,560, a silver requires 5,120, and the ultimate gold will need a score of 10,240.

     However, I wouldn’t suggest going for the Double or Nothing trophy at times other than the beginning of the month. The statistics don’t work out in your favor. Winning Round 7 is a 1/128 chance (.008%), Round 8 is a 1/256 chance (.004%), and Round 9 is a 1/512 chance (.002%). And to win Round 10, a pot of 10,240 NP? You’ve only got a 1/1,024 probability, which is a .001% chance. That means it may take over a thousand games to get a Round 10 win. This is possible if you have the patience for it…but I cannot recommend it!

     

 
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