So here is another post about probability and turning the odds in your favor.
So the concept of Penney's game is all about binary sequence. In the game two players chose their own sequence of heads and tails, usually 3 in length, and then flip a fair coin. The person whose sequence is generated first wins the game.
This is an example of a non-transitive game. This means that "one element in relation to a second element and the second in relation to a third element implies the first element is in relation to the third element, as the relation “less than or equal to.”." (http://dictionary.reference.com/browse/nontransitive).
So let's say you pick HHH and I pick THH. Who has I greater chance of winning? That would be me.
What if your sequence was HHT and mine was THH. Winner: me.
How about if your sequence is THH and mine is TTH. Again the winner would be me.
Why?
Let's go back to your first choice. HHH
what is the probability that you will get that? (1/2)^3=1/8. So 1/8 of the time you will win.
And my chances 1-1/8=7/8.
Meaning: I have a 7 to 1 chance of beating you.
Next: HHT
Now we'll use something called summation notation.
So the concept of Penney's game is all about binary sequence. In the game two players chose their own sequence of heads and tails, usually 3 in length, and then flip a fair coin. The person whose sequence is generated first wins the game.
This is an example of a non-transitive game. This means that "one element in relation to a second element and the second in relation to a third element implies the first element is in relation to the third element, as the relation “less than or equal to.”." (http://dictionary.reference.com/browse/nontransitive).
So let's say you pick HHH and I pick THH. Who has I greater chance of winning? That would be me.
What if your sequence was HHT and mine was THH. Winner: me.
How about if your sequence is THH and mine is TTH. Again the winner would be me.
Why?
Let's go back to your first choice. HHH
what is the probability that you will get that? (1/2)^3=1/8. So 1/8 of the time you will win.
And my chances 1-1/8=7/8.
Meaning: I have a 7 to 1 chance of beating you.
Next: HHT
Now we'll use something called summation notation.
Getting HHT
first 3 flips--> (1/2)^3=1/8
first 4 flips-->(1/2)^4=1/16
first 5 flips-->(1/2)^5=1/32
and so on...
first 3 flips--> (1/2)^3=1/8
first 4 flips-->(1/2)^4=1/16
first 5 flips-->(1/2)^5=1/32
and so on...
Chance of me winning: 1-1/4= 3/4
I have a 3 to 1 chance of winning.
Strategy?
So to win this game, start with the opposite of the middle choice of your opponent's sequence followed by the first 2 choices of your opponent's sequence. This makes the chances of the 1st player getting the beginning of their sequence allow the 2nd player to finish their sequence. Ex) P1: HHT P2:THH Flips:HTHTHTTHHTHHH
http://jammnpeaches.blogspot.com/
https://plus.maths.org/content/os/issue55/features/nishiyama/index
I have a 3 to 1 chance of winning.
Strategy?
So to win this game, start with the opposite of the middle choice of your opponent's sequence followed by the first 2 choices of your opponent's sequence. This makes the chances of the 1st player getting the beginning of their sequence allow the 2nd player to finish their sequence. Ex) P1: HHT P2:THH Flips:HTHTHTTHHTHHH
http://jammnpeaches.blogspot.com/
https://plus.maths.org/content/os/issue55/features/nishiyama/index