Real MATHEMATICS – What are the chances?! #6

How much luck?

Some people talk about how useless mathematics is. Maybe not all but most of these people lack the ability to understand probability of events, and they don’t even have any clue on how much damage they get because of that. Oddly enough whomever you ask, people will tell you how confident they are about probability.

One of the greatest examples for this is a casino game called roulette. If you have been to a casino before, you already know what I am going to talk about. For those who have no idea what roulette is, it is a gambling game in which a ball is dropped on to a revolving wheel with numbered compartments, the players betting on the number at which the ball comes to rest. Numbers on the wheel are between 0 and 36. Out of 37 numbers; 18 of them are colored in red, another 18 are black and only 1 of them (zero) is neither.

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In roulette one of the things you can gamble is color of the number. You can choose either red or black. It is not that hard to realize that the probability of red or black number winning the round is equal to each other and it is 18/37.

Q: Let’s say you are sitting on a roulette table and you observe that the last eight winners were all red. What would you bet on the next round: Red or black?

I asked ten of my friends and got these answers: 2 reds, 6 blacks and 2 “whatever”. Strange thing is nine of them said (without me asking them) that they know the probability is same in each round. It means that their choices were made instinctly, not mathematically.

The answer should be “whatever” as the probabilities stay the same in each round. Previous rounds have no effect on the next round. Except your psychology…

Warning: Gambling is the mother of all evil. Don’t do it!

“Are you a gambler?”

It is a multiplayer game. For “Are you a gambler?”, all you need is a standard dice.

indir (12)

Rules:

  • Players roll the dice in turns.
  • Outcomes of the rolls are added together. Goal is to be the first player who passes 50.
  • Whenever a player rolls 6, round ends for that player. Player also loses all the points he/she got in that round.
  • Players can stop their round whenever you want; as long as they don’t roll 6.

For instance if player A rolls 4 in the first round, he/she will get to choose one of the following two options: Roll the dice for the second time or end the round and add earned points (4 in this case) to his/her total. Let’s say player A decided to roll the dice second time and got 5. Again there will be two options for player A: Roll the dice for the third time or end the round and add (4+5=9 in this case) earned points to his/her total.

What to learn from all this?

There is only one number you avoid to roll and that is 6. This means whenever you roll the dice there is 5/6 = 0,833… (in other words 83%) chance that you will survive your round. Surviving two rolls consecutively has (5/6)*(5/6) = 0,694… probability. It means your chances have dropped to below 70% just after two rolls.

Imagine that you rolled a dice for 10 times. The probability of getting a 6 gets higher. (Almost 60%)

Although if one takes each roll individually probability of avoiding 6 never changes: 5/6. When you consider cases individually, you miss the bigger picture.

Unlikely events turn into most likely events within time. For instance getting a 6 after 100 rolls is almost certain:

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99,99999879% you will get a 6.

Conclusion: If you roll a dice 100 times, it is almost certain that you will get a 6. But if you take a look at each roll, probability of getting a 6 is always 1/6. This is why unlikely event becomes the most likely event within time.

Then I ask you: When do you stop rolling while playing “Are you a gambler?”?

One wonders…

Let’s change “Are you a gambler?” a bit. Assume that:

  • You don’t have to avoid 6 anymore.
  • Whenever a player hits the total of 5, 10, 15, 20, 25, 30, 35, 40 or 45, player loses all of his/her points.

What kind of strategy would you use? What do you think is different from the original game?

M. Serkan Kalaycıoğlu