Real Mathematics – What are the chances?! #5

Youngest ones will approve this: It is usually youngest kid’s duty to run errands for home. This is why I was the minister of errands until I left home for college. Walking to our local market covered most of my daily duties. On an average day I was walking towards the market more than a few times. On top of that, my school was about 20 meters away from that market. I memorized every centimeter square of that street while growing up.

Actually I wasn’t bothered with this as I was able to create games in any kind of situation in my childhood. For example as I was walking down that street I dribbled with stones. I pretended that every little round stone is a foot ball and I was the famous French footballer Zinedine Zidane.

My favorite game was something I called “walking the line” which I still play by myself.

Walking the Line

In my old neighborhood, floor was tiled with these stones:

In the game of walking the line, goal is to step inside the boundaries of those stones. For every step that crossed the boundary I get -1 score as for each of the successful steps that landed inside the boundaries of the stone I got +1.

Q: Pick any step during a walk. What is the probability of that step being a successful one?

Assumptions:

  • Assume that the shape of the foot is a rectangle.
  • Let the foot has sizes 30×6 cm.
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  • Assume that the shape of the stone tiling is a square.
  • Let the square has size 60×60 cm.
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  • Assume that foot always has the same direction when it lands on the square:

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In order to take a successful step, foot can touch the boundary but never cross it. This actually means that the center of the foot (or the rectangle) must be inside a specific area.

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Center of the rectangle.

Let’s assume that the top of the rectangle touches the top side of the square. In this case rectangle’s center would be exactly 15 cm away from the side of the square:

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If this distance is less than 15 cm, it means that the rectangle crosses the boundary of the square:

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Although, when this distance is between 15 and 45 cm, rectangle is considered to be inside the boundary of the square:

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One can conclude the same for the lower side of the square.

It is also possible to observe that when the distance between the center of the rectangle and the boundary of the square is less than 3 cm, rectangle crosses the boundary of the square:

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Although, when this distance is between 3 and 57 cm, one can know that rectangle is inside the square:

Conclusion

Thus, the probability of taking a random step that is successful (inside the boundary of the square) on a squared-tiling, foot’s/rectangle’s center must be inside the following area S:

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Area of the square: 60*60 = 3600 cm2.

Area of S: 54*30 = 1620 cm2.

Probability of taking a successful step on a square:

1620/3600 = 0,45.

This means that for the given measurements it is possible to take a step that will not exceed the boundary of the squared-tiling 45% of the time.

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One wonders…

  1. Take a coin and a chess board. Flip the coin on the board. What is the probability of this coin landing inside one of the squares on the chess board?
  2. What are the sizes of the coin and the square when there is more than 50% chance of winning?

M. Serkan Kalaycıoğlu